IN  MEMORIAM 
FLORIAN  CAJORI 


THE 

AMERICAN 

PHILOSOPHICAL  ARITHMETIC 

DESIGNED  FOR  THE  USE  OP  ADVANCED  CLASSES 
IN 

SCHOOLS  AND  ACADEMIES; 


CONTAINING 

THE    ELEMENTARY  AND   THE    MORE    ADVANCED    PRINCIPLES    OF 

THE    SCIENCE    OF    NUMBERS,  AND    THEIR   APPLICATIONS 

TO    PRACTICAL    PURPOSES, 

TOOETHER 

WITH    CONCISE    AND    ANALYTIC    METHODS    OF    SOLUTION,    AND 

ABBREVIATED    METHODS    OF   COMPUTATION. 

BY 

JOHN  F.  STODDA.ED,  A.M., 

PRINCIPAL  OF  THB 

LANCASTER    COUNTY,    NORMAL    SCHOOL,    PA. 

AIJTHOR  OF 

*•  THE   JUVENILE   MENTAL,"    "  THE   AMERICAN   INTELLECTUAL,"    AND 
"  THE   practical"   ARITHMETICS,    "  READY   RECKONER,"   EtC. 


NEW    YORK: 
SHELDON,    BLAKEMAN  &  CO. 

IBXLADKLPHU,  UPPLN-COTT,  GRAMBO  Jk  CO.     BOSTON,  JOHN  P.  JEWETT  &  CO. 

BUFFALO,  PHIXXEY  k  CO.    CI.KAELAXD,  KXIGDT,  KING  k  CO.    a.XCIN- 

NATI,  APPU:GATE  k  CO.    ClKa.E\TliE,  A.  BEACH  i  CO.    CHICAGO, 

KEENK  k  BRO.     OT.  LOITS,  E.  K.  WOODWARD, 

AND  KETfH  k  WOODS. 


Entered  according  kt  Act  of  Congi-3ss,  in  the  year  IKM.  or 

JOHN    F.    STODDARD, 

It.   th«  Clerk's   Office  of   the  District  Court  of  the  United   States,   for  tha 

Southern  District  of  New  York. 


JT- 


PEEF ACE 


*♦» 


I  HAVE  attempted  in  tMs  work  to  present 
CLEAELY  and  CONCISELY,  all  those  important 
principles  dindi prop^erties  of  numbers,  wMcli  are 
necessary  to  a  full  comprehension  of  the  higher 
branches  of  Mathematics,  and  their  application 
to  practical  business  and  scientific  calculations. 
The  arrangement  and  illustration  of  these  prin- 
ciples will  enable  the  pupil  to  think  for  himself, 
independent  of  arbitrary  rules,  and  summon  to 
his  aid  on  the  instant  all  his  arithmetical 
knowledge.  This  work  treats  extensively  of 
mercantile  transactions,  interest,  trade  and 
bai-ter,  abbre^dated  arithmetical  calculations, 
<fec.,  and  contains  several  principles  not  here- 
tofore published,  of  extensive  and  practical 
application. 

A  Key  containing  additional  methods  of 
analysis,  and  answer's,  (if  deemed  indispensable,) 
will  be  prepared  for  the  use  of  Teachers. 

J.  F.  S. 

UNivzRsmr  of  Nokthern  Pennsylvaioa,  I 
July  Uh,  1853.  f 


Digitized  by  the  I ntferhet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/americanphilosopOOstodric 


STODDARD'S 
PHILOSOPHICAL  ARITHMETIC. 


CHAPTER  I. 


INTRODUCTORY   DEFINITIONS. — NOTATION. NUMERATION. 

Article  1.  Science  is  knowledge  systematized  ;  that 
is,  knowledge  so  classified  and  arranged  as  to  be  conve- 
niently taught,  easily  acquired  and  remembered,  readily 
referred  to,  and  advantageously  applied. 

Art.  2.  Art  is  a  judicious  application  of  science  to 
practical  purposes. 

Art.  3.  A  Unit,  or  Unity ^  is  a  single  thing  ;  as,  a  tree, 
an  apple,  a  boy,  &c. 

Art.  4.  A  Number  is  either  a  unit  or  composed  of  an 
assemblage  of  units.     One,  two,  three,  &c.,  are  numbers. 

Art.  5.  .Quantity  is  anything  that  will  admit  of  meas- 
urement. A  line,  a  surface,  time,  and  other  things  of  this 
nature,  are  quantities  :  but  imagination,  reason,  virtue, 
&c.,  are  not  quantities,  therefore,  they  are  not  subjects  of 
mathematical  investigation. 

In  common  language,  quantity  or  numbers  are  expressed 
by  the  words  07ie,  two,  three,  four,  &c.;  in  arithmetical 
operations,  by  characters  called  Figures. 

Art.  6.  Mathematics  is  the  science  that  treats  of  the 
properties  and  relations  of  quantity.  Its  fundamental 
branches  are  Arithmetic,  Algebra,  and  Geometry. 

Art.  7.  Arithmetic  is  the  science  of  numbers,  and  the 
art  of  computation  by  them.  It  treats  theoretically  and 
practically  of  the  nature  and  properties  of  numbers  as 
employed  in  calculation. 


6  NOTATION.  [chap.    I 

Art.  8.  Algebra  is  a  general  method  of  investigating 
the  relations  of  quantity,  by  means  of  letters  and  signs,  or 


Art  9.  Geometry  is  the  science  of  magnitude  ;  it  esti- 
mates and  compares  extension  and  form. 

Art.  10,  An  abstract  number  is  one  that  does  not 
refer  to  any  particular  denomination  ;  as,  one,  six,  ten, 
"•Jive  hundred,  &c. 

Art.  1  i.  A  concrete,  or  denominate  number  is  one  that 
refers  to  some  particular  thing  or  denomination;  as,"  four 
apples,  ten  dollars,  &c. 

Similar  concrete  numbers  express  the  same  kind  of  units; 
as,  ten  dollars  &ud  Jifty  dollars. 

Dissimilar  concrete  numbers  express  different  kinds  of 
units  ;    &sjlve  dollars,  ten  horses. 

NOTATION.       • 

Art.  1 2.  Notation  is  the  art  of  expressing  numbers  by 
figures,  letters,  or  other  symbols. 

The  Romans  used  the  seven  following .  letters  to  ex- 
press numbers,  which  we  now  use  to  number  Lessons, 
Chapters,  &c.  : — I,  for  one  ;  Y,  for  five  ;  X,  for  ten  ; 
L,  for  fifty;  C,  for  one  hundred  ;  D,  for  five  hundred  ; 
M,  for  one  thousand.  The  intermediate  numbers  and 
numbers  greater  than  a  thousand  are  expressed  by  repeti- 
tions and  combinations  of  these  letters,  as  exhibited  in  the 
following 

ROMAN   table. 

One  is  represented  by  I. 
Two  "  "II. 

Three  "  "   III.  As  often  as  a  letter  is  re- 

Four  "  "    IV.  peated,  its  value  is  repeated, 

pjyg  «  «    Y^  Thus  :  X,  is  ten  ,  XX,  twen 

Six  "        "  yi. 

Seven  "  "   YII. 

Eight  "  "   VIII. 

Niao  "  "  IX. 


ty,  &,c. 


ART.    14.] 

NOfATION. 

Ten  is  represented  by  X. 

Eleven 

(< 

XI. 

Twelve 

ti 

XII. 

Thirteen 

it 

XIII. 

Fourteen  * 

It 

XIY. 

Fifteen 

11 

XY. 

Sixteen 

(( 

XYI. 

Seventeen 

(( 

XYIL 

Eighteen 

ti 

XYIII. 

Nineteen 

11 

XIX. 

Twenty 

n 

XX. 

Thirty 

tl 

XXX. 

Forty 

n 

XT,. 

Fifty 

tl 

L. 

Sixty 

tl 

LX. 

Seventy 

tt 

LXX. 

Eighty 

tl 

LXXX 

Ninety 

It 

xc. 

One  hundred 

tt 

c. 

Five  hundred 

tt 

D. 

One  thousand 

tt 

M. 

Five  thousand 

tt 

Y. 

A  letter  of  less  value 
placed  before  one  of  greater, 
diminishes  its  value  as  much 
as  the  value  of  the  letter 
placed  there  ;  if  placed  after 
the  same  letter  it  increase? 
its  value  by  the  same  num- 
ber. Thus,  if  before  X,  ten, 
we  place  I,  one,  it  becomes 
IX,  nine ;  if  after,  it  be- 
comes XI,  eleven.  If  before 
L,  fifty,  we  place  X,  ten,  it 
becomes  XL,  forty  ;  if  after, 
it  becomee  LX,  sixty,  &c. 


A  bar  ( — )  placed  over 
any  letter  increases  its  value 
a  thousand  fold.  Thus,  IV 
is  four  thousand. 


ARABIC    NOTATION. 

Art.  13.  In  all  arithmetical  calculations  numbers  are 
expressed  by  the  Arabic  system  of  Notation.  This  system 
of  notation  employs  the  following  ten  characters,  called 
Figures  : 

1,       2,      3,        4,      5,      6,       1,-        8,         9,         0. 
one,  two,  three,  four,  five,  six,  seven,  eight,  nine,  cipher, 

zero,  or 
naught. 
Hence,  figures  are  representatives  of  numbers,  or  expres- 
sions of  quantity. 

Art.  14.  The  first  nine  of  the  above  characters,  are 
called  significant  figures,  as  each  one  represents  a  definite 
number  when  standing  alone,  while  the  last  has  no  value, 
and  is,  therefore,  of  itself  insignificant. 


8  NUMERATION.  [?HAP.    I. 

Art.  15.  The  significant  figures  are  also  called  Digits, 
from  the  Latin  digitus,  a  finger,  because,  many  centuriea 
ago,  people  used  to  do  their  reckoning  by  counting  their 
fingers.  The  use  of  the  ten  fingers  is  supposed  to  have 
originally  suggested  the  idea  of  employing  ten  characters 
to  express  numbers. 

Art.  16.  Notwithstanding  the  0,  cipher,  has  no  value 
of  itself,  yet  it  is  of  as  much  importance  as  either  of  the 
digits,  as  it  serves  in  a  peculiar  manner-  to  change  their 
value  by  changing  their  locality ;  hence,  by  some  it  has 
been  called  a  Locater. 

NUMERATION. 

SIMPLE    AND     LOCAL   VALUES    OF    FIGURES. 

Art.  17.  A  figure  standing  alone,  or  occupying  the 
first  place  on  the  right  of  a  row  of  figures,  expressing  a 
whole  number,  is  denominated  units  ;  that  occupying  the 
second  place,  tens  ;  that  occupying  the  third  place,  huii- 
dreds,  &c. 

Thus,  Hds.     Tens.    Units. 

4  2  8 
is  read  four  hundreds,  two  tens,  and  eight  units,  or  four 
hundred  and  twenty-eight.  Nine  is  the  largest  number 
that  can  be  expressed  by  a  single  figure  ;  hence,  numbers 
greater  than  nine  must  be  expressed  by  some  combination 
of  these  numerals. 

Art.  18.  The  simple  value  of  a  figure  is  its  value  when 
occupying  unit's  place. 

The  name  of  a  figure  expresses  the  -simple  value  of  that 
figure.  Each  of  the  nine  digits  as  referred  to  in  Art.  13, 
has  its  simple  value.  In  the  number  35,  the  five  has  its 
simple  value,  fii^e,  and  the  three  a  local  value. 

Art.  19.  The  local  value  of  a  figure  is  that  which 
arises  from  its  location.  In  the  number  35,  (above  re- 
ferred to,)  the  local  value  of  the  three  is  three  tens,  or 
thirty     In  the  number  465,  the  local  value  of  the  4  is 


ART.    20.]  NUMERATION.  9 

four  hundreds,  aud  that  of  the  6  is  six  tens,  or  sixty.    The 
5  has  its  simple  value,  Jive. 

Art.  20.  When  the  nine  digits  occupy  the  second,  or 
ten's  place,  each  will  then  express  the  same  number  of  tens 
that  it  did  units,  when  occupying  the  place  of  units.  When 
they  occupy  the  third,  or  hundreds'  place,  each  will  then 
express  as  many  hundreds  as  it  did  units  when  in  the 
place  of  units. 

Note. — This  will  be  rendered  plain  by  inspecting  the  following 
TABLE. 

H  ■ 

No     units  and  one  ten,  or  ten,        ^10 

One  unit  "  "  "  "  eltve7L,    ^11 

Two  units  "  "  "  "  tioelve,        12 

Three    "  "  "  "  "  thirteen,     13 

Four     "  "  "  "  "  fourtcm,    14 

Five     "  "  "  "  "  fifteen,       15. 

Six        "  "  "  "  "  sixteen,       16 

Seven   "  "  "  "  "  seventeen.    It 

Eight   "  "  "  "  "  eighteen,    18 

Nine     "  "  •'  "  "  nineteen,     19 

No        "  "  two  "  "  twenty,       20    ^ 

Note. — The  terms  thirteen,  fourteen,  fifteen,  sixteen,  &c.,  are  obviously 
contractions  of  three  and  ten,  four  and  ten,  five  and  ten,  six  and  ten,  &c. 

In  a  similar  way,  by  contracting  the  expressions  two  tens,  three  tens,  four 
tens,  &o.,  the  expressions  twenty,  thirty,  forty,  &c.,  are  derived. 

Twenty-one,  Twenty-two,  Twenty-three,  Twenty-four, 
Twenty-five,  Twenty-six,  Twenty-seven,  Twenty-eight,  and 
Twenty-nine,  are  respectively  expressed  by  .placing  in 
regular  order  the  digits  in  the  place  of  the  cipher 
in  the  number  20.  In  a  similar  manner,  numbers  from 
Thirty  to  Forty,  from  Forty  to  Fifty,  from  Fifty  to  Sixty, 
&c ,  are  expressed  by  placing  the  digits  in  the  place  of 
the  cipher  in  the  numoers  Thirty,  Forty,  Fifty,  Six* 
ty,  &c. 


10 


NUMERATION. 


fcHAP.    I. 


No     units     and 


three 

four 

five 

six 

seven 

eight 

nine 


tens,     or 


Thirty, 

Forty, 

Fifty, 

Sixty, 

Seventy, 

Eignty, 

Ninety, 


HP 
30 

40 
50 
60 
70 
80 
90 


The  terms  twenty-one.  twenty -two,  &c.,  are  comi)ounded  of  twenty  and  one, 
twenty  and  two,  &c.  Other  numbers  expressed  by  two  figures  are  similarly 
formed. 


One  hundred, 

Two  " 

Three  " 

Four  " 
Five 
Six 

Seven  " 

Eight  " 

Nine,  " 


100 
200 
300 
400 
500 
600 

too 

800 
900 


Art.  21.  By  inspecting  the  above  table,  it  will  be 
observed,  that  a  figure  standing  in  the  second  place,  or 
place  of  tens,  is  ten  times  as  great  as  though  it  were  in  the 
first  or  units'  place.  A  figure  that  stands  in  the  third 
place,  or  place  of  hundreds,  is  ten  times  as  great  as  though 
it  were  in  tens'  place,  and  one  hundred  times  as  great  as 
though  it  were  in  the  place  of  units.  Hence,  ten  units 
make  one  ten,  and  ten  tens  make  one  hundred.  We 
therefore  infer  universally,  that 

Art.  22.  Figures  increase  in  value  from  right  to  left  in 
a  tenfold  ratio  ;  that  is,  each  removal  of  a  figure,  ove  place 
towards  the  left  increases  its  value  ten  tim£s. 

Art.  23.  As  figures  in  the  Arabic  system  of  notation 
increase  in  ten-fold  ratio  from  right  to  left,  and  decrease 
in  the  same  ratio  in  an  opposite  'Erection,  it  is  called  the 
Decimal  system  of  notation.  The  ^ord  decimal  is  derived 
from  the  Latin,  lecem,  ten. 


ART,  24.]  KUMERATION.  11 

Art.  24,  Numeration  is  the  art  of  reading  numbers, 
expressed  by  fig-ures. 

Note. — By  carefully  studying  the  following  Table,  the  pupil  will  soon  he 
able  to  read  any  number  which  requires  not  more  than  nine  figures  to  ex- 
press it. 

TABLE. 


•1  C    ^.  1 


<U     M     3     ♦-  J3 


5    s 


_  Figures  occupying  the  place  of  units,  are 

i  ^  o  '^  S  -  sometimes  called  units  of  the  first  order, 

^  iS  °  2  2  — those  occupying  the  place  of  tens,  vnits 

3  2  2  '^  of  the  second  order, — those  accupying  the 

t  ^^  °  ^  place  of  hundreds,  units  of  the  third  order , 

~          ^  &c.,  as  shown  by  the  Table. 

S  -^  w-  -S  ^  "I  -3  r'  S  '^^  read  these  numbers,  first  denominate 

£o  g  S^o  |ti®°  each  figure  by  the  names  units,  tens,  &c., 

c  c  3  c  c  c  c  S  .^  as  shown  by  the  table,  and  then  read  from 

"  i  K  ^  H  =  H  J§  left  to  right  as  follows  :~ 

....  4    Four. 
...  4  5     Forty-five. 
..453    Four  hundred  and  fifty-three." 
.4532    Four  thousand  fi.ve  hundred  and  thirty- two 
4  5  3  2  6^  Forty-five  thousand  three  hundred  and  twen- 
l  ty-six. 
4  5  3^67^  Four  hundred  fifty-three  thousand  two  hun- 
(  dred  and  sixty-seven. 
4  5  32678^  Four  millions,  five  hundred  thirty-two  thou- 
(  sand,  six  hundred  and  seventy-eight. 
45326781^  Forty-five  millions,  three  hundred  twenty- 
\  six  thousand,  seven  hundred  and  eighty-one 
C  Four  hundred  fifty-three  millions,  two  hun- 
453267819<  dred   sixty-seven   thousand,    eight   hundred 
C  and  nineteen. 

Rkmark.— It  would  be  well  to  write  the  figures  of  this  Table  on  the  black 
board,  and  have  the  pupils  read  them  individually  as  well  as  collectively. 

This  Table  shows  plainly  the  simple  and  local  values  of  figures.  Each 
figure,  except  those  in  the  place  of  units,  has  a  local  value,  which  may  b« 
named  by  th«  pupil  an  th«  taach«r  points  to  tham  separataly. 


12 


NUMERATION. 


[CHAI.    I. 


Art.  25.  In  the  United  States  and  continental  Eu- 
rope the  French  method  of  numeration  is  in  general  use. 
In  this  method  of  numeration  a  different  name  is  given  to 
every  three  figuers,  counting  from  the  right. 

The  first  period  contains  units,  tens  of  units,  hundreds 
of  units,  and  is  therefore  called  the  period  of  Units.  For 
a  similar  reason  the  next  left  hand  period  is  called  the 
eriod  of  Thousands,  &c. 

Art.  26.  In  the  following  Talle  the  words  above  the 
row  of  figures  express  the  particular  denomination  of  the 
figures  over  which  they  are  placed. 

To  read  a  number  expressed  by  figures  : — 
Denominate  each  figure  from  right  to  left,  remembering  the 
name  of  each  period,  then  read  the  figures  of  each  period,  he- 
ginning  at  the  left  hand,  in  the  same  manner  as  those  of  the 
period  of  units  are  read,  and  at  the  end  of  each  period  give 
its  name. 

FRENCH    METHOD    OF   NUMERATION. 


S.2 

m  O; 


3    0)   4i 

7  i  3. 


=!      ^5      .E-^      2-   .^o      |=§      =. 


-:::        ><■ 


QfB 


o    •<— .S     .^ .°    .^7^    ■'^^zi    -t^c 

^    wo—;    "J'-Z:2     wO-rr     tnOu     »CO 


S3 .  s; 


o  c  « 


4,  6 


o  -'<u;r'3a>m  30)3   3(u3   3ais-'   3a)r;;  3 
3,  8  5  7,  8  6  2,  0  7  3,  4   1   0,  2  0  6,  3  7  4,  3 


"2  " 

£■« 

ilil 

;  S  E-i  r-' 
8  4  7. 


6  2. 


Period    Period    Period    Period    Period    Period    Period    Period    Period    Period    Period    Pericd 
of  De-    ofNo-    ofOc-    of  Sep-  of  Sex-  of  Quin-ofQuad- of  Tril    of  Bil-    of  Mil- of  Thou-       of 
cUliona.  nillions.  tillions.  tillions.  tillions.  tillions.  rillions.    lions,      lions,      lions.     t»nda.    UniU. 


EXERCISES  IN  NUMERATION. 


Kead  the  following  numbers  : — 


Ex. 

Ex. 

1. 

125 

6. 

2. 

286 

7. 

3. 

397 

8. 

4. 

8462 

9. 

5. 

98623 

10. 

Ex. 

63245 

11. 

836478492 

732123 

12. 

326489472 

8324671 

13. 

8246721478 

1247632 

14. 

448624783146 

463246273 j   15.       12345678901234 


ART.  28.]  FUNDAMENTAL   RULES.  13 

EXERCISES  IN  NOTATION. 

Art.  27.  To  express  numbers  by  figures  . — 

Begin  at  the  left,  and  write  the  figures  of  the  highest 
order  mentioned,  observing  to  place  in  each  order,  the  figures 
belonging  to  it,  and  ichen  no  digit  is  mentioned,  to  fill  the 
flace  with  a  cipher. 

Express  the  following  numbers  by  figures  : — 

1.  Forty-three. 

2.  Eighty-nine. 

3.  Three  hundred  and  eight. 

4.  Four  thousand,  one  hundred  and  four. 

5.  Seventy-five  thousand  and  seventy-five. 

6.  Six  hundred  and  five  thousand,  one  hundred  and 
twenty-three. 

7.  Eight  hundred  and  seventy-two  thousand,  five  hun- 
dred and  twelve. 

8.  Nine  millions,  seven  hundred  and  sixty-five  thousand, 
four  hundred  and  thirty-two. 

9.  Three  hundred  and  forty  millions,  forty-three  thou 
sand,  five  hundred  and  sixty-seven. 

10.  Three  hundred  and  seventy-four  billions,  four  hundred 
and  thirty-eight  millions,  eight  hundred  and  sixty-two 
thousand,  eight  hundred  and  forty-seven. 

FUNDAMENTAL  RULES  OF  ARITHMETIC.  . 

Art.  28.  JVotation  aud  IVumeration  are  the  Frimarj  -pnn- 
ciples  of  the  four  Fundamental  llules  of  arithmetic;  namely, 
Addition,  Subtraction,  Multiplication,' and  Division. 

These  are  called  Fundamental  Rides,  because  all  other 
arithmetical  operations  are  dependent  on  them. 

A  Rule,  in  Arithmetic,  is  a  prescribed  method  of  per- 
forming an  Arithmetical  operation. 


14  ADDITION.  [chap.    )l. 


CHAPTER  II. 

ADDITION, SUBTRACTION. MULTIPLICATION. DIVISION. 

AUDITION. 

Art.  29.  Addition  is  the  process  of  finding  the  sum 
of  two  or  more  numbers. 

The  sign  of  addition  is  a  shori  horizontal  line  bisected 
by  a  perpendicular  line  of  the  same  length;  as,  +•  This 
symbol  is  called  plus,  and  when  placed  between  two  quan- 
tities, it  denotes  that  they  are  to  be  added.  Thus,  4  +  2 
show  that  four  and  two  are  to  be  added;  and  is  read,/oi^r 
plus  two. 

Two  parallel  horizontal  lines,  as  =,  means  equal  to  or 
eqiials,  and  when  placed  between  two  quantites,  denotes 
that  they  are  equal  to  each  other.  Thus,  4 -f  2  =  6,  is  read, 
4  plus  2  equals  6. 

CASE  I. 

Art.  30.  Addition  of  abstract  numbers  when  the  sum 
of  each  column  does  not  exceed  nine. 

Rkimark. — Similar  concrete  Mumbers  are  added  the  same  as  though  they  were 
abstract  numbers,  the  amount  being  a  concrete  number  of  the  same  kind. 
Disiimilar  concrete  numbers  cannot  be  added. 

1.  What  is  the  sum  of  223,  451  and  114  ? 

Explanation.  — Write  the  numbers 
OPERATION.  to  be  added,  so  that  ;dl  the  figures  of 

^  the  same  denomination  shall  stand  in 

^  ^^  m     '  the  same  column;  and  draw  a  line 

§  g  3  underneath.      Then    commencing   at 

^  ^  ^  the  column  of  units,  add  each  column 

^  ,  ^  separately,  and  phice  the  result  direct- 

^  ^  ^  ly  under  it.     Thus,  4  and   1  are  5, 

■*■  and  3  are  S. units,  which   place  under 

_       „~  the  column  of  units.    Add  the  column 

7  8  8  Sum  or  Amount.     ^^^^^^^^  ^^^  of  hundreds  in  a  similar  way, 
and  we  obtain  for  the  amount  788. 
Proof. — Begin  at  the  top  and  add  eacli  column  down- 
ward, the  same  as  you  added  them  upward,  if  the  sunaa 
ajjreo  the  work  is  right. 


T.  30.J 

ADDITION. 

2.      3. 

4. 

5. 

6. 

7. 

134     131 

171 

315 

413 

112 

512     413 

403 

481 

142 

221 

342     245 

315 

202 

234 

'   343 

8. 

9. 

10. 

1142 

4131 

10465 

3314 

1512 

43110 

2001 

3024 

22322 

1330 

201 

13101 

15 


11.  What  is  the  sum  of  3141,1202  and  2382  ? 

12.  What  is  the  sum  of  1674,3102,4011  and  112  ? 

13.  What  is  the  sum  of  12132,41311,23323  and  1101? 

14.  What  is  the  sum  of  3421,30124,313221  and  1232  ? 

15.  What    is    the    sum    of    1210  +  32124  +  613253  + 
2110301  +  2101? 

16.  What    is    the    sum    of     1012  +  32421  +  613352  + 
2110103  +  2011  ? 

17.  What  is  the  sum  of  3121  +  21  +  1603  +  1032  ?, 

18.  What  is  the  sum  of  413  +  32132  +  32  +  4220  ? 

19.  What  is  the  sum  of  12  +  3430  +  40  +  64213  +  104  ? 

20.  What  is  the  sum  of  12  +  321  +  4231  +  821423  +  12? 


PRACTICAL    QUESTIONS. 

1.  If  a  yoke  of  oxen  is  worth  125  dollars,  and  a  cow  32 
dollars,  what  is  the  value  of  both  ? 

2.  A  man  bought  a  load  of  hay  for  5  dollars,  a  load  of 
wheat  for  41  dollars,  and  some  rye  for  323  dollars;  what 
was  the  w^hole  cost  ? 

3.  A  farmer  bought  a  span  of  horses  for  212  dollars,  a 
yoke  of  oxen  for  132  dollars,  and  farming  implements  to 
the  amount  of  545  dollars;  what  was  the  whole  cost  ? 

4.  A  merchant  sold  2131  barrels  of  flour  one  month, 
11023  barrels  the  next  month,  and  6022  barrels  the  follow- 
ing month  ;  how  many  barrels  did  he  sell  during  the  three 
months  ? 


16  ADDITION.  [chap.    II. 

5.  A  man  bought  some  butter  for  123  dollars,  some 
molasses  for  310  dollars,  some  sugar  for  1101  dollars,  and 
some  flour  for  1121  dollars;  what  was  the^  whole  cost  ? 

6.  James  has  2312  acres  of  land,  John  has  21321,  and 
Joseph  has  32154;  how  many  acres  have  they  all  ? 

t.  A  farmer,  being  asked  how  many  sheep  he  had,  re- 
plied, in  one  field  I  have  225,  in  another  2112,  in  another 
1220,  and  in  another  10120;  how  many  had  he  in  all  ? 

8.  A  farmer  raises  the  following  qunatities  of  grain  oa 
four  fields,  namely  :  on  the  first  2115  bushels  of  wheat,  on 
t^e  second  1110  bushels  of  rye,  on  the  third  625  bushels 
of  oats,  and  on  the  fourth  123  bushels  of  buckwheat; 
how  many  bushels  of  grain  did  he  raise  ? 

9.  A  lends  to  B  1313  dollars  ;  to  C  23121  dollars,  and 
has  55125  dollars  remaining  ; — how  much  money  had  A 
at  first  ? 

10.  A  man  bought  a  farm  for  4120  dollars,  paid  1400 
dollars  for  having  it  improved,  and  sold  it  so  as  to  gain 
1150  dollars  ;  for  how  much  did  he  sell  it  ? 

-  11.  A  man  traveled  214  miles  one  day,  232  miles  the 
next  day,  and  320  the  third  day  ;  how  far  did  he  travel  in 
the  three  days  ? 

12.  Mr.  Smith  owned  five  farms  ;  the  first  was  wof-th 
23520  dollars,  the  second  11120  d-ollars,  the  third  3200 
dollars,  and  the  fourth  32100  dollars  ;  what  is  the  value  of 
the  four  farms  ? 

13.  A  drover  bought  cattle  to  the  amount  of  3100 
dollars,  sheep  to  the  amount  of  642  dollars,  and  a  fine 
horse  for  255  dollars  ;  how  much  did  they  all  cost  ? 

14.  A  merchant  bought  groceries  to  the  amount  of  3210 
dollars,  dry  goods  to  the  amount  of  12210  dollars,  and  had 
32216  dollars  remaining  ;  how  much  had  he  at  first  ? 

15.  A  had  2310  dollars,  B  13250  dollars,  C  32118  dol- 
lars, and  D  321  dollars  ;  how  much  did  they  together 
have  ? 

16.  A  merchant,  on  setthng  up  his  business,  found  he 
owed  one  man  12326  dollars,  another  412  dollars,  another 
3141  dollars,  another  821010  dollars  ;  what  was  the 
amount  of  his  debts  ? 


ART.    31.]  ADDITION.  It 

11.  A  has  2113413  dollars  ;  B,  534231  dollars  ;  C,  343 
dollars  ;  and  D,  85241002  dollars  ; — how  many  dollars 
have  they  together  ? 

18.  In  one  book  there  are  1210  pages,  in  another  235, 
and  in  another  1140;  how  many  pages  did  the  three  books 
contain  ? 

19.  A  merchant  bought  books  to  the  amount  of  1111 
dollars,  paper  to  the   amount  of  2231  dollars,  and   dry 

•goods  to  the  amount  of  23225  dollars;  how  much  did  the 
whole  cost  ? 

20.  A  man  has  a  farm  worth  1522  dollars,  a  mortgage 
worth  23134  dollars,  and  5222  dollars  of  bank  sock;  how 
much  rs  he  worth  ? 

CASE    II. 

Art.  3 1 .  Addition  of  abstract  numbers  in  general. 

1.  What  is  the  sum  of  431t  +  346  +  59  +  6831  +  2194 
+  3285? 


Explanation. — Write  the  numbers  to  be 
added  as  directed  in  Art.  30. 

Begin  at  units'  column  and  add  thus  :  5 
and  4  are  9,  and  1  is  10,  and  9  are  19,  and 
6  are  25,  and  7  are  32  units, — equal  to  3 
TENS  and  2  units ; — place  the  2  units  under 
the  units'  column,  and  car-y,  or  add,  the  3 
tens  to  the  tens'  column,  thus :  3  and  8  are 
11,  and  9  are  20,  and  3  are  23,  and  5  are 
28,  and  4  are  32,  and  2  are  34  tens, — equal 
to  3  HUNDREDS  and  4  tens ; — place  the  4 
tens  under  the  tens'  column  and  add  the  3 
hundreds  to  the  hundreds'  column,  thus  :  3 
and  2  are  5,  and  1  is  6,  and  8  are  14,  and  3 
are  17,  and  3  are  20,  hundreds, — equal  to  2 
THOUSANDS  and  0  hundreds ; — place  the  0  hundreds  under  the 
hundreds'  column  and  add  the  2  thousands  to  the  column  of 
thousands,  thus  :  2  and  3  are  5,  and  2  are  7,  and  6  are  13,  and 
4  are  17  thousands, — equal  to  1  ten  thousand  and  7 thousands, 
*- place  the  7  thousands  under  the  column  of  thousands,  and 


operation. 

1 
1 

Thousands. 
Hundreds. 
Tens. 
Units. 

H 

4327 

"3 

346 

?, 

59 

6831 

2194 

3285 

17042  Amount. 

18  ADDITION.  [chap.    H. 

the  1  ten  thousand  on  the  left  of  it ;  and  we    have  for  the 
amount  17042. 

Proof  hy  the  excess  of  9'5. — Find  the  excess  of  9's  in  the 
sum  of  the  digits  of  each  of  the  numbers  added,  and  if 
the  ejxcess  of  9's  in  these  excesses,  equals  the  excess  of  9's 
in  the  product,  the  work  may  be  considered  right. 

Take  for  illustration  the  preceding  example. 

OPERATION. 

4327  =  7  excess. 
346  =4  " 
59  =  5  " 
6831  =0  " 
2194  =  7  '^ 
3285  =  0      " 


Amount,  17042  =  5,  is  the  excess  of  9's  in  the  above 

hence  the  work  is 


right. 


Rkmark. — To  comprehend  this  method  of  proof,  as  well  as  that  given  for  the 
proof  of  Subtraction,  Multi{>lication  and  Division,  it  is  necessary  to  under- 
stand the  {)roperties  of  the  number  9  explained  on  page  80th,  Art.  74. 


2. 

3. 

4. 

5. 

6. 

3412 

7310 

782 

3241 

18243 

3410 

416 

4164 

476 

32341 

218 

32 

3123 

84324 

7147 

436 

4 

7182 

18472 

165 

1412 

74- 

119 

31421 

2342 

7.  What  is  the  sum  of  4862+834+46734  +  82796  f 
9832     8763? 

8.  What  is  the  sum   of  144  +  7864  +  891234  6327  + 
9879? 

9.  What  is  the  sum  of  78639  +  847796+864321  +  1487 
+  987? 

10.  What  is  the  sum  of  186  +  ^72+4638  +  64732  + 
8634  +  9763+478? 


ART.    31.]  ADDITION-.  19 

11.  Find  the  sura  of  986+834  +  t325t  +  t63244  8t63 
4-9876  +  4683  +  9824. 

12.  Find  the  sum  of  8632  +  84129  +  91  +  1864  +  9981  + 
1632  +  876324. 

13.  Find  the  sura  of  11468+3121  +  863+4902+816  + 
8196+81641  +  163. 

14.  What  is  the  sum  of  8463  +  121  +  84632+8468  + 
1416+8916+868411  +  9816141  ? 

•  15.  What  is  the  sura  of  846832  +  981649+168321  + 
684  +  9163+84162  +  9824  ? 

16.  What  is  tlie  sura  of  18461  +  982+6849+131241  + 
6824121  +  168411  +  9163+4214  ? 

11.  What  is  the  sum  of  16824+4168+4134+8686  + 
9432  +  981  +  98624? 

18.  What  is  the  sum  of  23416+1862541+91632+8163 
+9168+92+8416+1231  ? 

19.  What  is  the  sum  of  86432+68324  +  981324+83241 
+  964? 

20.  What  is  the  sura  of  16841+9683+8324+8632+  , 
149118+16832+1984683+86432141  ? 

PRACTICAL    QUESTIONS. 

1.  A  father  gave  to  his  eldest  son  1413  dollars,  to  his 
youngest  son  3249  dollars,  to  his  oldest  daughter  1298 
dollars,  to  his  youngest  daughter- 3998  dollars,  and  had 
remaining  1968  dollars  ; — how  much  money  had  he  at 
first  ? 

2.  Several  persons  contributed  towards  building  a 
church.  A  gave  184  dollars,  B  gave  213  dollars,  C  gave 
843  dollars,  D  gave  195  dollars,  and  E  gave  395  dol- 
lars ;— how  much  did  they  together  contribute  ? 

3.  Five  brothers  had  the  following  sums  of  money  ;  A 
9189  dollars,  B  15450  dollars,  C  899  dollars,  D  3499  dol- 
lars, and  E  9999  dollars  ; — how  much  did  they  together 
have  ? 

4.  A  drover  bought  491  sheep  one  week,  841  the  next 
week,  943  the  third  week,  1496  the  fourth  week,  and  18550 
the  fifth  week  ; — how  many  sheep  did  he  buy  in  all  ? 


20  ADDITION.  [chap.    II. 

5.  A  gentleman  owns  a  farm  wortli  3450  dollars,  a  build- 
ing lot  worth  3759  dollars,  a  store  and  lot  worth  5868 
dollars,  a  fine  horse  and  carriage  worth  715  dollars  ; 
what  is  the  amount  of  his  property  ? 

6.  From  New  York  to  Kingston  is  90  miles,  from  King- 
ston to  Albany  is  60  miles,  from  Albany  to  Rochester  is 
251  miles,  from  Rochester  to  Buffalo  is  75  miles,  and  from 
Buffalo  to  Niagara  Falls  is  21  miles  ;  how  far  is  it  from 
New  York  to  Niagara  Falls  ? 

7.  An  individual  owns  a  farm  worth  2463  dollars,  a 
wood-lot  worth  1342  dollars,  a  store  and  lot  worth  2465 
dollars  ;  what  is  the  amount  of  his  property  ? 

8.  A  gentleman  willed  his  estate  to  his  wife,  three  sons, 
and  four  daughters  ;  to  his  daughters  he  willed  3496  dol- 
lars apiece;  to  his  sons,  each  5785  a  piece;  and  to  his  wife 
4698  dollars  ; — how  much  was  his  estate  ? 

9.  The  distance  on  the  New  York  and  Erie  railroad 
from  New  York  to  Goshen  is  59  miles  ;  from  Goshen  to 
Narrowsburgh  is  63  miles  ;  from  Narrowsburgh  to  Owego 
is  114  miles  ;  from  Owego  to  Friendship  is  137  miles  ;  and 
from  Friendship  to  Dunkirk  is  87  miles.  How  many  miles 
from  New  York  to  Dunkirk  ? 

10.  A  boy  gave  for  a  slate  22  cents  ;  for  an  arithmetic 
50  cents  ;  for  an  algebra  75  cents ;  for  a  grammar  56 
cents;  and  for  a  geography  125  cents.  How  much  did  he 
give  for  them  all  ? 

11.  A  butcher  sold  to  one  man  436  pounds  of  meat  ; 
to  another  3695  pounds  ;  to  another  9899  pounds  ;  to 
another  12485  pounds  ;  and  to  another  879  pounds.  How 
many  pounds  did  he  sell  in  all  ? 

12.  A,  B,  C,  D  and  E  enter  into  partnership  ;  A  puts 
in  475  dollars  ;  B  846  dollars  ;  C  1495  dollars  ;  D  985 
dollars  ;  and  E  7864  dollars.  How  much  stock  have  they 
in  trade  ? 

13.  Four  persons  deposit  money  in  a  bank  ;  the  first 
deposits  4490  dollars  ;  the  second  5685  dollars  ;  the  third 
9947  dollars  ;  and  the  fourth  12470  dollars.  How  many 
dollars  did  they  all  deposit  ? 

14.  Bought  of  A  346  cords  of  wood  ;  of  B  846  cords  ; 


ART.    31.]  ADDITION.  21 

of  C  395  cords  ;  of  D  836  cords  ;  of  E  as  much  as  of  A 
and  C  both  ;  and  of  F  as  much  as  of  B  and  E  both. 
How  many  cords  of  wood  did  I  buy  in  all  ? 

15.  A  produce-dealer  has  in  store  at  one  place  146 
bushels  of  corn,  876  bushels  of  oats,  395  of  rye,  and  1247 
bushels  of  potatoes  ;  at  another  place  1846  laushels  of 
corn,  3246  bushels  of  oats,  846  bushels  of  rye,  and  437 
bushels  of  potatoes  ;  and  at  another  place  199  bushels  of 
corn,  847  bushels  of  oats,  and  849  bushels  of  potatoes ;— 
how  much  produce  has  he  in  store  .'' 

16.  Macedon  was  founded  794  years  B.  C.  by  Caranus  ; 
Sparta  was  founded  606  years  before  Macedon,  by  Selex  ; 
Corinth,  4  years  before  Sparta,  by  Lysippus  ;  Thebes,  89 
years  before  Corinth,  by  Cadmus.  In  what  year  was 
Sparta,  Corinth,  and  Thebes  founded  respectively  ? 

17.  The  population  of  the  United  States  in  1790  was 
3729326  ;  in  1800  it  was  1580427  more  ;  1810  it  had  in- 
creased 1930150  more  ;  in  1820,  2398377  more  ;  in  1830, 
3218241  more  ;  and  in  1840,  4244165  more.  What  was 
the  population  in  each  of  the  above  mentioned  years  ? 

18.  Mr.  Harvey,  the  discoverer  of  the  circulation  of 
the  blood,  was  born  in  1578,  at  Folkstone,  in  Kent ;  George 
Edwards,  the  ornithologist,  was  born  116  years  later  ; 
William  Herschel,  the  astronomer,  was  born  44  years  after 
Edwards ;  Henry  Clay,  the  American  statesman,  was 
born  39  years  after  Herschel; — in  what  year  was  each 
of  the  above  named  individuals  born  ? 

19.  At  the  battle  of  Moskowa  there  were  13000  Rus- 
sians killed,  5000  taken  prisoners,  about  27000  wounded, 
and  40  generals  either  killed,  wounded  or  taken  prisoners; 
2500  of  Napoleon's  army  were  killed,  7500  wounded, 
and  15  generals  either  killed  or  wounded.  What  was 
the  total  loss  ? 

20.  At  the  battle  of  Waterloo  the  French  lost  40000 
men;  the  Prussians  38000;  the  Belgians  and  Dutch  8000; 
the  Hanoverians  3500;  and  the  English  about  12000; — ■ 
how  many  men  were  killed  in  all  ? 


23  SUBTRACTION.  [CHAP.  II 

SUBTRACTION. 

Art.  32.  Subtraction  is  the  method  of  finding  the  dif- 
ference between  two  numbers. 

In  subtraction  there  are  three  terms,  the  Mimiend,  Sub- 
trahend, and  Remninde?- .  Any  two  of  these  being  given, 
the  remaining  one  can  be  found. 

The  number  from  which  the  other  is  to  be  taken  is 
called  the  Minuend;  the  number  to  be  subtracted  from  it, 
the  Suhtraheiid;  and  the  result  obtained  by  the  operation, 
the  Remainder. 

A  short  horizontal  line,  thus,  — ,  is  called  minus,  and  is 
the  sign  of  suMraction,  When  it  is  placed  between  two 
numbers,  it  shows  ihat  the  number  on  the  right  of  it  is  to 
be  taken  from  the  one  on  the  left.  Thus,  t,  (the  minuend) 
—  5,  (the  subtrahend)  =  2,  the  remainder,. 

CASE    I. 

Art.  33.  Subtraction  of  abstract  numbers,  when  each 
figure  of  the  subtrahend  is  less  than  its  corresponding 
figure  in  the  minuend. 

Remark. — The  diflerence  of  two  similar  concrete  numbers  is  a  concrete 
number  of  the  same  kind,  and  is  found  in  the  same  way  as  though  they  were 
abstract  numbers.  But  two  dissimilar  concrete  numbers  can  not  be  taken, 
the  one  from  the  other. 

1.  From  946  subtract  524. 

Explanation. — Write  the  less  num- 
operation.  ber  under  the  greater,  with   units  unv 

^  der  units  &c.,   and  draw  a  linfe  under- 

"i  neath.  then  proceed  thus :  4  units  from 

6  units,  leave  2  units  :  vrrite  the  2  units 
in  units'  place,  2  tens  from  4  tens  leave 
,..  J  c\)ia  2  tens,  vehich  write  in  tens'   place,  5 

^'u^^l'  A      lol  hundreds   from   9    hundreds   leavQ    4 

.Subtrahend,     bZi  hundreds :  write  the    4    hundreds  in 

,,        .J  Ann         the  place  of  hundreds,  and  we  have 

Kemamder,       4  2  2         ^^^^  ^{J^  ,emainder  422. 

Proof. — Add  the  remainder  and  subtrahend  together;  if 
their  sum  is  equal  to  the  minuend,  the  work  is  right. 


p:  0-  « 


ART.    33.] 


SUBTRCTIOS. 


38 


2. 

From         465 
Subtract    243 


From 
Subtract 


3. 

842 
511 


4. 
From         762 
Subtract    451 


5. 
From  549 
Subtract     334 


From 
Subtract 


6. 
465 
143 


7. 
From  947 

Subtract     837 


8.  From  4631  take  2310. 

9.  From  16820  take  3410. 

10.  From  9642  take  8431. 

11.  From  32478  take  12374. 

12.  From  96472  take  32361. 

13.  Subtract  4247  from  7449. 

14.  Subtract  147302  from  688925. 

15.  Subtract  234610  from  479824. 

16.  From  9867412  subtract  4243101. 

17.  From  1649324  subtract  443121. 

18.  From  256342  subtract  143242. 

19.  From  9864324  subtract  8432124. 

20.  From  9680434  subtract  45304U. 


PRACTICAL  QUESTIONS. 

1.  A  boy  had  36  marbles  and  gave  24  of  them  to  his 
playmate;  how  many  had  he  remaining  ? 

2.  Joseph  caught  295  quails,  and  John  caught  84;  how 
many  more  did  Joseph  catch  than  John  ? 

3.  Jackson  had  95  cents  and  Jane  had  73;  how  many 
more  had  Jackson  than  Jane  ? 

4.  Elisha  having  447  bushels  of  potatoes,  sold  2S4 
bushels  of  them  to  Perry;  how  many  bushels  had  he 
remaining  ? 

5.  A  farmer  bought  a  span  of  horses  for  346  dollars,  a 
yoke  of  oxen  for  135  dollars;  how  much  more  did  he 
give  for  the  horses  than  for  the  oxen  ? 

6.  A  drover,  having  1465  sheep,  sold  1235  of  them  ; 
how  many  had  he  remaining  ? 


24  SUBTRACTION.  [cHAP.    II. 

I.  A  gentleman  owns  a  store  worth  4695  dollars,  and  a 
grist-mill  worth  2135  dollars  :  how  much  more  is  the  store 
worth  than  the  grist-mill  ? 

8.  A  gentleman  gave  for  a  house  and  lot-  9899  dollars, 
for  a  cotton  factory  8495  dollars  ;  how  much  more  did  he 
give  for  the  one  than  the  other  ? 

9.  A  speculator  bought  some  land  for  1289t  dollars,  a 
tannery  for  10444  dollars  ;  how  much  more  did  the  land 
cost  than  the  tannery  ? 

10.  A  merchant,  having  9841  yards  of  cloth,  sold  5844 
yards  of  it ;  how  many  yards  had  he  remaining  ? 

II.  A  drover  bought  cattle  to  the  amount  of  9647  dol- 
lars, and  sheep  to  the  amount  of  5434  dollars  ;  how  much 
more  did  he  give  for  the  cattle  than  for  the  sheep  ? 

12.  A  merchant  sold  a  quantity  of  goods  for  869*7  dol- 
lars, and  by  so  doing  gained  1495  dollars  ;  how  much  did 
the  goods  cost  him  ? 

13.  A  gentleman  sold  an  estate  for  1499  dollars,  and  by 
so  doing  gained  1084  dollars;  how  much  did  the  estate 
cost  him  ? 

'     14.  A  farm  was  sold  for  3495  dollars,  which  was  1032 
dollars  more  than  it  was  worth;  how 'much  was  it  worth  ? 

15.  A  farmer  had  4295  sheep,  and  2145  lambs  ;  how 
many  more  sheep  had  he  than  lambs  ?. 

16.  A  farmer,  having  1346  bushels  of  wheat,  sold  1042 
bushels  of  it  ;  how  many  bushels  had  he  remaining  ? 

11.  A  merchant,  during  one  year,  sold  1241  barrels  of 
molasses,  and  2489  barrels  of  sugar  ;  how  many  more 
barrels  of  sugar  did  he  sell  than  molasses  ? 

18.  A  gentleman  willed  to  his  son  49865  dollars,  and  to 
his  daughter  34534  dollars  ;  how  much  more  did  he  will  to 
his  son  than  to  his  daughter } 

19.  A  man,  driving  1565  sheep  to  market,  on  his  way 
sold  435  of  them  ;  how  many  had  he  remaining  ? 

20.  A  ship  is  valued  at  69841  dollars,  and  its  cargo  at 
45831  dollars  ;  how  much  more  is  the  ship  valued  at  than 
the  cargo  ? 


ART.  34.1  SUBTRACTION.  25 


I  CASE    II. 

Art.  34.  Subtraction  of  Abstract  nambers  in  geteral. 
1.  From  728  subtract  364. 

OPERATION.  Explanation.  —  The  numbers 

being  properly  written  down,  we 

•S  -S  proceed  thus  :  4  units  taken  from 

•«  •  2    -a     «    i  °  units  leave  4  units,  which  write 

o  §  a     3     g    '3  in  unit's  place.     I  cannot  take  6 

K/f-         ^        nn^    ?  /i^N  o  tens  from  2  tens  ;  therefore,  from 

^IT?^'  ^    n  5=^  ^^^^  ^  the  7  hundreds  I  take  1  hundred 
Subtrahend,  36^  ^  ^q  ^^^^^  ^^^  ^^^  i^  ^^  ^^^^  2 

T,        .   J         oo  A  tens,   making    12    ^ews,- — 6   tens 

Remamder,    3  6  4  ^^^^  12  tens  leave  6  tens,  which 

I  write  in  tens'  place.  I  have 
taken  1  hundred  from  the  seven  hundreds,  which  leaves  6  hun- 
dreds; 3  hundreds  from  6  hundreds  leave  3  hundreds.  But  for 
convenience,  it  is  customary  to  add  the  1  hundred  to  the  3 
hundreds,  (the  next  figure  in  the  subtrahend.)  and  take  the 
sum  from  the  figure  in  the  minuend  under  which  it  is  placed, 
which  is  the  same  in  effect  as  the  above. 

Note. — The  minuend  7  hundreds,  2  tens  and  8  units,  is  =  6  hundreds,  12 
t£ns  and  8  units,  which  form  it,  absolutely  assumes  in  the  mind  while  giving 
the  above  explanation;  still  it  is  not  necessary  to  be  written  except  to  render 
the  explanation  more  plain. 

Proof  by  the  excess  of  9'5. — Find  the  excess  of  9's  in  the 
sum  of  the  digits  of  the  remainder, — also  of  the  subtrahend. 
Then  find  the  excess  of  9's  in  the  excesses  just  found, — 
if  this  excess  equals  the  excess  of  9's  in  the  vtinucnd,  the 
work  is  right. 

Take  for  illustration  the  above  example  : 

OPERATION. 

728  =  8  excess. 
364  =  4      " 

364  =  4      « 


8  excess  in  the  subtrahend  and 
remainder,  which  is  the  same  as  the  excess  in  the  minuend, 
therefore  titie  work  is  right. 

2 


26  SUBTRACTION.                                  [CHAP.  II 

2.  3         4.        5. 

From   4642  From  647   From  4621  From  468 

Take   2370  Take  352   Take  2432  Take  379 


fi  7  R 

From  68492       From    *    7246    From       68243 

Subtract      37508      Subtract  5839    Subtract  27359 


9.  From  8697  subtract  5988. 

10.  From  1682402  subtract  740482. 

11.  From  187642  subtract  94837. 

12.  From  9046  subtract  8074. 

13.  From  86432  subtract  67821. 

14.  Subtract  4962  from  7832. 

15.  Subtract  14829  from  84643. 

16.  From  4001  subtract  1344. 

OPERATION.  2nd.  Zrd.  4tli, 


m  Xi 


■T3  m 


Minuend,       4001  =  3(10)01  =  39    (10)1=399(11) 
Subtrahend,  1344  1344 


Remainder,     2657  2657 

Explanation. — The  numbers  being  properly  arranged,  com 
mence  at  the  right  and  proceed  thus  :  we  cannot  take  4  units 
from  1  unit ;  therefore  I  seek  1  from  the  tens'  place,  but  find- 
ing no  tens  there,  I  proceed  to  the  hundreds'  place,  and  finding 
no  hundreds  there,  I  take  1  thousand  from  the  4  thousands  and 
r  set  it  in  the  next  place  towards  the  right,  which  causes  the 
minuend  to  take  the  2nd  form.  Then  take  1  hundred  from  the 
10  hundreds,  and  set  it  in  the  next  place  towards  the  right, 
causes  the  minuend  to  take  the  3rd  form.  Now  taking  1  ten 
from  the  10  tens,  and  adding  it  to  the  1  unit  causes  the  minuend 
to  assume  the  4th  form, — from  which  we  are  now  prepared  to 


A.RT.  34.]  SUBTRACTION.  2*1 

take  the  subtrahend.  4  units  from  11  units  leave  7  units ;  4 
tens  from  9  tens  leave  5  tens ;  3  hundreds  from  9  hundreds 
leave  6  hundreds ;  and  1  thousand  from  3  thousands  leave  2 
thousands.  Hence  the  difference  of  these  two  numbers  is  2657. 

Remark.— The  2nd,  3rd  and  4th  forms  of  the  minuend  serve  merely  to  ex- 
plain the  method  of  subtracting  more  clearly,  and  should,  therefore,  in  practice, 
be  performed  in  the  mind  and  not  be  written  down.  The  sanffe  result  will  be 
obtained  by  simply  adding  10  to  the  upper  figure  when  it  is  smaller  than  the 
one  below  it,  and  carrying,  or  adding,  1  to  the  next  figure  of  the  subtrahend. 

ir  From  41007  subtract  34138  ? 

18.  From  90006  subtract  9994  ? 

19.  How  many  are  10000—9  ? 

20.  How  many  are  100000—1  ? 

21.  How  many  are  89467—84732  ? 

22.  How  many  are  760743—249078  ? 

23.  How  many  are  4078603—1437908  ? 

24.  How  many  are  90807060—60708091  ? 

25.  How  many  are  97876757—79787675  ? 

26.  How  many  are  20304050—1020304  ? 

27.  How  many  are  90857565—20382468  ? 

28.  How  many  are  900000—1  ? 

29.  How  many  are  909090—1  ? 

30.  How  many  are  9080706050—16070809? 

PRACTICAL    QUESTIONS. 

1.  A  gentleman  willed  to  his  son  3862  dollars,  and  to  his 
daughter  5324  dollars;  how  much  more  did  he  will  to  his 
daughter  than  to  his  son  ? 

2.  In  a  certain  orchard  there  are  425  apple-trees  and 
297  plum-trees;  how  many  more  apple-trees  than  plum- 
trees  ? 

3.  A  man  traveled  14637  miles  during  one  year,  and 
9843  miles  the  next  year  ;  how  much  farther  did  he  travel 
the  first  year  than  the  second  ? 

4.  A  merchant  had  25694  pounds  of  pork,  and  sold 
19832  pounds  of  it;  how  many  pounds  remained  unsold  ? 

5.  A  speculator  bought  a  quantity  of  cotton  for  294682 
di)llars,  and  sold  it  for  516390  dollars;  how  much  did  he 
gain  ? 


28  SUBTRACTION.  [CHAP.  II. 

6.  Gunpowder  was  invented  by  Schwartz,  in  the  year 
1330;  how  long  was  it  before  the  birth  of  Bonaparte, 
1769? 

I.  George  Washington  died  in  the  year  1*199,  at  the 
age  of  6t;  in  what  year  was  he  born  ? 

8.  The  mariner's  compass  was  invented  at  Naples  in  the 
year  1302;  how  long  before  the  discovery  of  America  1492? 

9.  Joseph  Addison,  the  poet,  was  born  16t2,  and  died, 
111 9;  how  old  was  he  when  he  died  ? 

10.  Sir  William  Blackstone,  the  lawyer,  was  born  1123, 
and  died,  1180;  at  what  age  did  he  die  ? 

II.  Francis  Bacon,  a  universal  genius,  died  in  the  year 
1626,  at  the  age  of  65;  in  what  year  was  he  born  ? 

12.  Robert  Burns,  the  poet,  was  born  1*159,  and  died 
1196;  Lord  Byron,  the  poet,  was  born  lt88,  and  died 
1824.  What  was  the  age  of  each,  and  how  long  after  the 
birth  of  Burns  was  Byron  born  ? 

13.  George  Edwards,  the  ornithologist,  was  born  1694; 
how  long  was  this  before  the  birth  of  Harvey,  the  dis- 
coverer of  the  circulation  of  the  blood,  who  was  born 
1.578  ? 

14.  Massachusetts  was  settled  in  1620,  at  Plymouth; 
how  many  years  before  the  declaration  of  our  National  In- 
dependence 1776  ? 

15.  The  Independence  of  the  United  States  was  ac- 
knowledged in  Europe  in  1783;  how  long  was  that  after 
the  battle  of  Bunker's  Hill,  1775  ? 

16.  The  first  newspaper  published  in  America,  at  Bos- 
ton, was  in  1704,  which  was  183  years  after  Mexico  was 
conquered  by  the  Spaniards;  in  what  year  was  Mexico 
conquered  ? 

17.  Michael  Angelo,  an  Italian  painter,  died  1568,  at 
the  age  of  89;  in  what  year  was  he  born  ? 

18.  Benjamin  Franklin,  the  philosopher  and  statesman, 
died  1790,  at  the  age  of  84;  in  what  year  was  he  born  ? 

■    19.  Galileo,  an  Italian  astronomer,  died  1642,  at  the 
age  of  78;  in  what  year  was  he  born  ? 

20.  Luther,  the  reformer,  died  1546,  at  the  age  of  63, 
in  what  year  was  he  born  ? 


ART.  34.]  SUBTRACTION.  29 

21.  Raphael,  the  prince  of  painters,  an  Italian,  was 
born  1483,  and  died  in  1520,  which  was  6  years  after  the 
birth  of  Titian,  another  renowned  Italian  painter;  to  what 
age  did  Raphael  live,  and  in  what  year  was  Titian  born  ? 

22.  Cotopaxi,  the  highest  volcano  in  the  world,  is  19408 
feet  high;  how  much  higher  is  Sorato,  the  highest  land  in 
America,  which  is  25380  feet  high,  than  Cotopaxi  ? 

23.  Benjamin  West,  the  American  painter,  was  bora 
1*138;  how  long  was  this  before  the  death  of  Robert  Ful- 
ton, who  died  in  the  year  1815  ? 

24.  Mount  Ararat,  (on  which  Noah's  ark  rested,)  is 
12100  feet  high;  now  how  much  higher  is  that  than  mount 
Washington  in  New  Hampshire,  which  is  6234  feet  in 
height  ? 

25.  St.  Peter^s  Church  at  Rome,  is  450  feet  high;  how 
much  higher  is  that  than  Trinity  Church,  New  York, 
which  is  283  feet  in  height  ? 

26.  Joseph  Bonaparte  died  1844,  at  the  age  of  16  ;  in 
what  year  was  he  born  ? 

2T.  Dr.  Franklin  was  born  in  the  year  1106,  and  died 
in  1190;  how  old  was  he  when  he  died  ? 

28.  A  man,  owning  45161  acres  of  land,  sold  23921 
acres  of  it;  how  many  acres  had  he  remaining  ? 

29.  A  merchant,  having  98012  barrels  of  flour,  sold 
49261  of  them;  how  many  had  he  remaining  ? 

30.  In  a  certain  town  there  were  24961  inhabitants, 
which  was  5084  more  than  there  were  the  preceding  year; 
how  many  were  there  the  preceding  year  ? 

31.  A  merchant  sold  a  quantity  of  goods  for  38961 
dollars,  which  was  813  dollars  more  than  they  cost  him; 
how  much  did  they  cost  him  ? 

32.  A  man,  having  21695  feet  of  lumber,  sold  1962  feet 
of  it;  how  many  feet  had  he  remaining  ? 

33.  If  I  borrow  of  my  neighbor  9613  dollars,  and  pay 
him  999  dollars  of  it;  how  much  remains  unpaid  ? 

34.  A  gentleman  sold  a  farm  for  54623  dollars,  which 
was  9240  dollars  more  than  he  gave  for  it;  how  much  did 
he  pay  for  the  farm  ? 

35.  A  farmer  raised  2141  bushels  of  rye,   and  2146 


30  SUBTRACTION.  [CHAP.    II. 

bushels  of  corn;  he  sold  943  bushels   of  the  rye,  and  189 
bushels  of  the  corn ; — how  much  of  it  remains  unsold  ? 

36.  A  and  B  bought  a  farm  for  7840  dollars;  A  paid 
2999  dollars,  and  B  the  remainder; — how  many  dollars 
did  B  pay  ? 

37.  A  and  B  traded  farms;  A's  farm  is  valued  at  9863 
dollars,  and  B's  at  7807  dollars; — how  much  in  equity 
ought  B  to  pay  A  ? 

38.  Said  A  to  B,  I  have  4605  sheep;  B  replied,  that 
he  had  as  many,  lacking  298; — how  many  had  B  ? 

39.  How  many  years  from  1496,  the  year  in  which 
Algebra  was  first  known  in  Europe;  to  1808,  the  year  in 
which  the  first  steamboat  was  put  in  successful  operation 
by  Robert  Fulton  ? 

40.  A  grocer  having  346823  dollars'  worth  of  goods, 
shipped  196832  dollars'  worth  of  them;  how  many  dollars' 
worth  had  he  remaining  ? 

41.  A  speculator  sold  a  factory  for  35896  dollars, 
which  was  1491  dollars  more  than  it  cost  him;  how  much 
did  it  cost  him  ? 

PRACTICAL  QUESTIONS  COMBINING  ADDITION  AND    SUBTRACTION. 

1.  A  farmer;  having  4632  sheep,  sold  to  A  785,  and  to 
B  896;  how  many  had  he  remaining  ? 

2.  A  farmer's  yearly  income  was  1679  dollars;  he  paid 
for  repairing  his  house  487  dollars;  for  farming  utensils 
98  dollars;  and  for  hired  help  299  dollars; — how  much 
has  he  remaining  ? 

3.  A  man  bought  a  span  of  horses  and  a  wagon  for 
987  dollars;  he  then  sold  the  wagon  for  185  dollars,  and 
the  horses  for  736  dollars; — how  much  did  he  lose  by  the 
operation  ? 

4.  A  gentleman,  having  697  dollars,  deposited  372 
dollars  in  the  bank,  and  spent  197  dollars  of  it;  how  much 
had  he  remaining  ? 

5.  A  speculator,  having  346821  acres  of  land,  sold  to 
A  637  acres;  to  B  495;  to  C  1865;  toD  26942;  and  to  E 
879  acres; — how  many  acres  had  he  left  ? 


ART.    34.]  SUBTRACTION.  31 

6.  There  is  a  farm  consisting  of  946  acres;  35  acres  of 
which  is  planted  with  corn  and  potatoes;  140  acres  sown 
with  rye;  180  acres  with  oats;  98  with  wheat;  212  is 
pastured,  and  the  remainder  is  meadow.  How  many  acres 
of  meadow  ? 

7.  A  lady,  having  467  dollars,  paid  for  a  bonnet  24 
dollars;  for  a  shawl  85  dollars;  for  a  silk  dress  90  dollars; 
and  for  some  delaines  112  dollars; — how  much  had  she 
remaining  ? 

8.  A  market-woman,  having  234  oranges,  sold  to  one 
person  12  of  them;  to  another  46;  to  another  54;  to 
another  32;  and  to  another  15; — how  many  had  she  re- 
maining ? 

9.  A  farmer,  having  89*1  sheep,  sold  to  A  150  of  them; 
to  B  160;  to  C  284;  and  to  D  294;— how  many  had  he 
remaining  ? 

10.  A  drover,  having  191  cattle,  sold  112  of  them,  and 
bought  81  more;  how  many  had  he  then  ? 

11.  In  a  certain  army  there  are  4560  men:  in  a  battle 
646  of  them  were  killed,  49t  of  them  wounded,  and  148 
of  them  deserted;  how  many  were  left  ? 

12.  A  farmer,  having  847  bushels  of  grain,  sold  to  A 
132  bushels;  to  B  112;  to  C  184;  and  gave  to  the  poor 
212  bushels; — how  many  bushels  had  he  remaining  ? 

13.  An  individual  traveled  by  railroad  497  miles,  and 
designed  to  return  on  foot;  the  first  day  he  traveled  69 
miles;  the  second  84;  the  third  59;  the  fourth  47  miles; 
the  fifth  day  he  took  the  cars  and  arrived  home.  How 
far  did  he  go  the  last  day  ? 

14.  A  man  willed  an  estate  of  560048  dollars  to  his 
two  children  and  wife,  as  follows  :  to  his  son  230645 
dollars;  to  his  daughter  88999  dollars;  and  to  his  wife  the 
remainder.     How  much  did  he  will  to  his  wife  ? 

15.  A  man  laid  out  98000  dollars  in  speculation;  the 
first  year  he  gained  1847  dollars;  the  second  year  1987 
dollars;  the  third  year  he  lost  8044  dollars.  How  much 
did  he  lose  by  the  operation  ? 

16.  A  merchant,  having  89776  barrels  of  flour,  sold  to 
A  967  barrels;  to  B  1743  barrels;  to  C  6842  barrels; 


32  MULTIPLICATION.  [CHAP.    II. 

to  D  14625  barrels;  and  to  E  the  remainder.     How  many 
barrels  did  E  receive  ? 

IT.  Four  persons  A,  B,  C  and  D  propose  to  purchase 
a  manufactory,  valued  at  97802  dollars.  A  is  to  pay 
4990  dollars,  B  1264Wollars,  C  19682  dollars,  and  D 
the  remainder;  what  sum  will  D  have  to  pay  ? 

18.  Having  in  my  possession  8960  dollars,  I  wish  to 
know  how  much  I  must  add  to  this  sum,  to  be  able  to 
purchase  a  farm  worth  18910  dollars,  and  save  497  dol- 
lars for  other  purposes  ? 

19.  A  had  448  oxen;  B  had  212  more  than  A;  and  0 
had  as  many  as  A  and  B  together,  lacking  184; — how 
many  had  B  and  0  respectively  ? 

20.  A  has  470  dollars  more  than  B,  and  245  dollars 
less  than  C,  who  has  2490  dollars;  and  D  has  as  much  as 
A  and  B  together.  How  many  dollars  have  A,  B  and  D 
respectively  ? 

21.  John  has  240  sheep  more  than  Joseph,  and  125 
less  than  James,  who  has  485 ;  and  Jackson  has  as  many 
as  John  and  Joseph  together,  lacking  320  sheep.  How 
many  sheep  have  John,  Joseph  and  Jackson  respect- 
ively ? 

MULTIPLICATION. 

Art.  35.  Multiplication  is  a  concise  method  of  com- 
puting  the  amount  of  any  number  taken  as  many  times  as 
there  are  units  in  another  number. 

There  are  three  terms  employed  in  multiplication;  the 
Multiplicand,  the  Multiplier,  and  the  Product;  any  two  of 
which  being  given  the  remaining  one  can  be  found. 

Art.  36.  The  MultplicaTid  is  the  number  taken. 
The  Multiplier  is  the  number  that  shows  how  many  times 
the  multiplicand  is  taken.  The  Product  is  the  answer  or 
result  obtained.  The  multiplicand  and  multiplier  are  also 
called  Factors  of  the  product. 

Art.  37.  The  multij^lier  can  never  be  a  concrete  num- 


XRT.  38.] 


MULTIPLICATION. 


88 


ber,  as  it  merely  expresses  the  number  of  times  the  multi- 
plicand is  taken.  The  product  will  be  of  the  same  denomi- 
nation as  the  multiplicand. 

Art.  38.  The  sign  of  multiplication  is  two  short  lines  of 
equal  length  bisecting  each  other  at  an  angle  of  45  de- 
grees with  the  horizon;  thus,  X,  and  is  sometimes  called 
INTO.  This  sign  being  placed  between  two  numbers  shows 
that  they  are  to  be  multiplied,  the  one  by  the  other.  Thus, 
6x8=48,  indicates  that  6  is  to  be  multplied  by  8,  or  8 
to  be  multiplied  by  6,  (as  the  case  may  require,)  and  that 
the  product  equals  48. 

MULTIPLICATION  TABLE. 


2X 
2X 
2X 
2X 
2X 


0=  0 
1=  2 
2=  4 
3=  6 
4=  8 
2X  5=10 
2x  6=12 
2X  7=14 
2x  8=16 
2x  9=18 
2X10=20 
2X11=22 
2X12=24 


3X  0=  0 

3X 

3X 


1=  3 

2=  6 


3X  3=  9 


3X 
3X 


4=12 
5=15 


3X  6=18 


3X 
3X 
3X 


7=21 
8=24 


4X 


4X  1= 

4X  2= 

4X  3= 

4X  4= 

4X  5= 

4X  6= 

4X  7= 

4X  8= 

4X  9= 
4X10= 


0|5X 
4  5X 


3X10=30 

3X11=33:4X11 

3X12=3614X12 


5X 
5X 
5X 
5X 
5X 
5X 
5X 
5X 
5X 
5X 
5X 


0=  0 

1=  5 

2=10 

3=15 

4=20 

5=25 

6=30 

7=35 

8=40 

9=45 

10=50 

11=55 

12=60 


6X  0= 


1=  6 
2=12 
3=18 
6X  4=24 
6X  5=30 
6=36 
7=42 
8=48 
6x  9=54 
6X10=60 
6X11=66 
6X12=72 


6X 
6X 
6X 


6X 
6X 
6X 


0=  0 

1=  7 
2=14 
3=21 
4=28 
5=35 
6=42 
7=49 
8=56 
9=63 
fO 


/X 

7X 

7X 

7X 

7X 

7X 

7X 

7X 

7X 

7X 

7X10 

7X11=77 

7X12=84 


8X  0=  0 
8X  1=  8 

8X  2=16 
8X  3=24 
8X  4=32 
8X  5=40 
8X  6=48 
X  7=56 
8X  8=64 
8X  9=72 
8X10=80 
8X11=88 
8X12=96 


9X  0=    0 


9X  1= 

9X  2= 

9X  3= 

9X  4= 

9X  5= 

9X  6= 


9X 
9X  8^ 
9X  9: 
9X10: 

9X11: 

9X12: 


7= 


10  X 
lOX 
lOX 
lOX 
lOX 
lOX 
lOx 
10  X 
lOX 
lOx 
lOX 
10  X 
lOX 


0=    0 
1=  10 

2=  20 

3=  30 

4=  40 

5=  50 

6=  60 

7=  70 

8=  80 

9=  90 

10=100 

11=110 

12=120 


llX 
llX  1 
IIX  2 
llX 
llX 
llX 
llX 
llX 

11  X  8: 
11  X  9: 
11X10: 

11X11: 

111X12: 


0= 


12  X  0= 
12X  1= 
12  X  2= 
12  X  3= 
12X  4= 
12X  5: 
12X  6: 
12X  7= 
12  X  8= 
12  X  9. 
110  I  12X10= 
121  12x11. 
132  !  12X12= 


99 


=  0 
=  12 
:  24 

:  36 

=  48 

:  60 
:  72 
:  84 
:  96 
:108 
:120 
:132 
:144 


34  MULTIPLICATION.  [cHAP.  II. 

CASE  I. 

Art.  39.  Multiplicatimi  of  abstract  numbers,  when  the 
multiplier  does  not  exceed  9. 

I.  Multiply  846  by  8. 

OPERATION.  Explanation — Write  the  numbers  down, 

^  placing  units  under  units  ;  then  proceed 

c-5  from  right  to  left:  thus,  8  times  6  units 

S-S  ^  =j     are  48  units,  or  4  tens  and  8  units  ;. — place 

J  J  g  3     the  8  units  in  units'  place,  and  reserve  the 

Tv/r  1  •  V       J  o^'i     ^  ^^^  *®  add  to  the  next  product,  8   times 

Mu  tiplicand,8  464  tens  are  32  tens,  and  4  tens  added  are  36 

Multiplier,  8     ^g^g^  ^j.  3  hundreds  and  6  tens  ; — place  the 

T.    J  2      TT     ^  *^°^   ^^    *^°^'  place  and  reserve  the  3 

Product,     6  7  6  8     hundreds  to  add  to  the  next  product    8 

times  8  hundreds  are  64  hundreds,   and  3  hundreds  added  are 

67  hundreds,  or  6  thousands  and  7  hundreds,  which  write  down ; 

and  we  have  for  the  product  6768. 

Proof. — Multiply  the  multiplier  by  the  multiplicand;  if 
the  product  thus  obtained,  equals  the  first  product  the 
work  is  presumed  to  be  right. 

2.  Multiply  348  by  2. 

3.  Multiply  483  by  3. 

4.  Multiply  684  by  4. 

5.  Multiply  6482  by  4. 

6.  Multiply  14682  by  5. 
1.  Multiply  18623  by  6. 

,,  8.  Multiply  38943  by  T. 
9.  Multiply  28462  by  8. 
10.  Multiply  8946  by  T. 

II.  Multiply  7683  by  6. 

12.  Multiply  9898  by  9. 

13.  Multiply  6847  by  3. 

14.  Multiply  94762  by  6. 

15.  Multiply  88992  by  7. 

16.  Multiply  33449  by  8. 

17.  Multiply  884682  by  9. 

18.  Multiply  99999  by  5. 

19.  Multiply  897654  by  7. 

20.  Multiply  123456789  by  8. 


ART.   39. J  MTLTIPLICATION.  85 

TRACTICAL  QUESTIONS. 

1.  A  man  solvl  105  sheep,  at  3  dollars  a  piece;  how  much 
did  he  receive  for  iliera  ? 

2.  What  cost  184  barrels  of  flour,  at  6  dollars  a  barrel  ? 

3.  What  cost  198T  ncres  of  land,  at  9  dollars  an  acre  ? 

4.  What  cost  4t8C  barrels  of  sugar,  at  9  dollars  a 
barrel  ? 

5.  In  1  mile  there  are  5l^S0  feet;  how  many  feet  in  5 
miles  ?  '^ 

6.  In  1  mile  there  are  1160  yards;  how  many  yards  in 
5  miles  ? 

I.  If  9  men  can  mow  a  certain  moadow  in  18  days  ;  in 
how  many  days  can  one  man  do  the  same  ? 

8.  If  6  masons  cq^i  build  a  certain  \v:ill  in  149  days  ; 
in  how  many  days  can  one  mason  build  the  same  wall  ? 

9.  If  460  bushels  of  oats  will  feed  1  horse  11  months; 
how  many  bushels  will  be  required  to  feed  8  horses  the 
same  time  ? 

10.  Bought  245  cords  of  wood,  at  t  dollars  a  cord. 
What  did  the  whole  cost  ? 

II.  A  farmer  sold  8  horses,  at  253  dollars  a  piece  ;  how 
many  dollars  did  he  receive  for  them  ? 

12.  A  lady  bought  189  yards  of  ribbon,  at  6  cents  a 
yard;  how  much  did  it  all  cost  her  ? 

18.  What  cost  1786  boxes  of  raisins,  at  3  dollars  a 
box? 

14.  If  a  steamship  can  go  395  miles  in  1  day,  how  far 
can  she  go  in  9  days  ? 

15.  A  merchant  bought  2864  hats,  at  4  dollars  a  piece; 
how  much  did  he  pay  for  them  all  ? 

16.  A  farmer  sold  9  fat  oxen,  at  185  dollars  a  piece; 
how  much  did  he  receive  for  them  all  ? 

17.  In  1  day  there  are  1440  minutes  ;  how  many  min- 
utes in  8  days  ? 

18.  In  1  day  there  are  86400  seconds;  how  many 
seconds  in  5  days  ? 

19.  If  one  man  receive  4  dollars  a  week,  how  much 
will  an  army  of  35680  men  receive  in  6  weeks  ? 


36  MULTIPLICATION.  [CHAP.  II. 

20.  At  2  dollars  a  day,  each;  how  much  will  it  cost,  to 
board  685  men  1  days  ? 

CASE  II. 

Art.  40.  Multiplication  of  abstract  numbers  in  general  ? 
1.  Multiply  437  by  56. 

OPERATION.      Explanation. — Write  the  numbers  down  so  that 
^  units   stand  under   units,   tens  under   tens,    &c. 

^    J  „•  w     Begin  at  the  right  and  proceed,  thus, — 6  times  7 
§  g  '3-^    units  are  42  units,  or  4  tens  and  2  units  ;  write  the 
WHt"     2  units  in  units'  place,  and  reserve  the  4  tens  to 
4  3  7     add  to  the  next  product.     6  times  3  tens  are  18 
5  6     tens,  and  4  tens  added  are  22  tens,   or  2  hundreds 

and  2  tens^  &c.     We  next  multiply  by  the  5  tens. 

2  6  2  2  For  convenience  we  say  5  times  7  are  35,  and  place 
218  5         the  5  under  the  multiplier,  5,  that  is  in  tens'  place, 

and  reserve  the  three  hundreds  to  add  to  the  next 

2  4  4  7  2  product,  &c.  But  instead  of  5  times  7,  &c.,  it  is, 
50  times  7  units  =  350  units,  or  3  hundreds^  5  tens  and  0  units. 
I  therefore  placed  the  5  tens  in  tens'  place,  where  you  perceive 
it  belongs.  We  proceed  iifthe  same  way  to  explain  why  we 
place  the  right  hand  figure  of  the  product  in  the  third,  or 
hundreds'  place  when  multiplying  by  that  figure,  &c. 

Proof  hy  the  excess  of  9'^. — Find  the  excess  of  9's  in 
each  FACTOR.  Then  if  the  excess  of  9's  in  the  product  of 
these  excesses,  equals  the  excess  of  9's  in  the  product  of  the 
two  factors,  the  work  is  right. 

Take  for  illustration  the  preceding  example  ; 


Factors, 


OPERATION. 

437  =  5    excess. 
56  =  2 


2622        1  ,  excess  in  the  product  of  the  excesses. 
2185 


Product,      24472  =  1  '  excesses  in  the  product  of  the  factors. 

Explanation Commence  at  the  left  of  the  first  factor,  (the 

multiplicand,)  and  add,  thus,  4-j-3-f-7  are  14,  or  1  nine  and  5 


ART.    40.]  MULTIPLICATION.  31 

units,  the  excess  of  9's  in  that  factor.  Add  the  second  factor, 
(the  multiplier,)  5-f-6  are  11,  or  1  nine  and  2  units,  the  ex- 
cess of  9's  in  that  factor.  The  product  of  these  excesses  is  10, 
or  1  nine  and  1  unit,  the  excess  of  9's  in  the  product  of  the 
excesses.  Add  the  total  product,  2-|-4-f-4+7-|-2  are  19,  or  2 
nines  and  1,  the  excess  of  9's  in  the  product  of  the  factors. 
This  excess  equals  the  required  excess;  hence  the  work  is 
right. 

2.  Multiply  4624  by  35.  >^ 

3.  Multiply  3846- by  39. 

4.  Multiply  8462  by  47. 
6.  Multiply  7846  by  147. 

6.  Multiply  3976  by  183. 

7.  Multiply  2243  by  144. 

8.  Multiply  2882  by  414. 

9.  Multiply  1414  by  323. 

10.  Multiply  2463  by  382. 

11.  Multiply  8632  by  132. 

12.  Multiply  4862  by  897. 

13.  Multiply  9876  by  678. 

14.  Multiply  4567  by  7654. 

15.  Multiply  1234  by  4321. 

16.  Multiply  8362  by  8496. 

17.  Multiply  146832  by  8376. 

18.  Multiply  36847  by  8324. 

19.  Multiply  1384697  by  476324. 

20.  Multiply  897654321  by  123456789. 


PRACTICAL    QUESTIONS. 

1.  If  15  men  can  build  a  certain  wall  in  235  days,  how 
long  will  it  take  1  man  to  do  it  ? 

2.  If  45  men  can  accomplish  a  certain  piece  of  work 
in  360  days  by  working  8  hours  a  day,  how  many  days 
will  it  take  one  man  to  do  the  same  by  working  4  hours  a 
day  ? 

3.  If  360  bushels  of  oats  will  last  185  horses  3  days, 
how  long  will  it  last  1  horse  ? 


88  MULTIPLICATION.  [CHAP.    II. 

4.  A  drover  bonglitr  685  oxea  at  104  dollars  a  piece; 
what  was  the  cost  of  all  of  them  ? 

5.  A  merchant  bought  25  pieces  of  broadcloth,  each 
piece  containing  48  yards,  at  9  dollars  a  yard.  How 
much  did  he  pay  for  the  whole  ? 

6.  If  a  steamship  can  sail  18  miles  in  1  hour,  how  far 
can  she  sail  in  34  days  of  24  hours  each  ? 

I.  A  speculator  bought  8968  acres  of  land,  at  195  dol- 
lars an  acre.     How  much  did  the  whole  cost  him  } 

8.  In  1  furlong  there  are  660  feet;  how  many  feet  in 
8  furlongs,  (1  mile)? 

9  How  many  pounds  of  flour  are  there  in  395  barrels; 
there  being  196  pounds  in  each  barrel  ? 

10.  What  is  the  value  of  346^shares  of  railroad  stock, 
at  125  dollars  a  share  ? 

II.  How  many  pages  are  there  in  5896  books,  there 
being  394  pages  in  each  book  ? 

12.  A  speculator  bought  302  cattle,  and  293  times  as 
many  sheep;  how  many  sheep  did  he  buy  ? 

13.  If  a  garrison  of  men  consume  98*T  pounds  of  beef 
in  1  day;  how  many  pounds  will  a  garrison  containing 
twice  as  many  men  consume  in  365  days  ? 

14.  Farmer  A  has  245  acres,  sowed  with  wheat,  which 
produces  32  bushels  to  the  acre.  Farmer  B  has  360 
acres,  sowed  with  wheat,  which  produces  25  bushels  to 
the  acre.  What  quantity  of  wheat  was  raised  by  A  and 
B  respectively  ? 

15.  A  speculator  bought  146  head  of  oxen;  230  head 
of  cows;  and  69  head  of  calves.  He  made  a  profit  of  16 
dollars  a  head  on  the  oxen;  12  on  the  cows;  and  5  on  the 
calves.  How  much  did  he  gain  on  the  oxen,  cows,  and 
calves  respectively  ? 

16.  A  merchant  bought  12  boxes  of  linen,  each  con- 
'taining  25  pieces,  and  each  piece  containing  36  yards,  at 
65  cents  a  yard.  How  many  pieces,  and  how  many  yards 
did  he  buy,  and  how  much  did  it  all  cost  him  ? 

17.  A  has  395  acres  of  land,  worth  2t  dollars  an  acre; 
and  B  has  493  acres,  worth  19  dollars  an  acre.  What  is 
the  value  of  each  of  their  farms  ? 


4.RT.    41.  J  MULTIPLICATION.  89 

18.  In  a  certain  orchard  there  are  26  rows  of  apple- 
trees  and  36  trees  in  each  row.  How  many  apples  would 
there  be  in  the  orchard,  allowing  2595  apples  to  each  tree  ? 

19.  A  farmer  purchased  five  tracts  of  land,  each  con- 
taining 395  acres,  at  95  dollars  an  acre.  What  was  the 
whole  cost  ? 

20.  The  circumference  of  the  earth  is  nearly  25000 
miles,  the  distance  to  the  sun  is  3800  times  as  much.  What 
is  the  distance  to  the  sun  ? 

Art.  41.  A  Composite  number  is  one  that  can  be  pro- 
duced by  multiplying  two  or  more  numbers  together,  each 
of  which  is  greater  than  a  unit. 

Thus,  15  is  a  composite  number,  as  it  can  be  produced 
by  multiplying  together  the  numbers  3  and  5.  The  3  and 
5  are  called  the  factors  of  15. 

Art.  42.  When  the  multiplier  is  a  composite  number^, 
resolve  it  into  two  or  more  factors,  then  multiply  the  mul- 
tiplicand by  one  of  these  factors,  and  the  product  thus 
obtained  by  another  factor,  and  so  on,  until  all  the  fac- 
tors have  been  used  as  a  multiplier.  The  last  product 
will  be  the  answer  sought 

1.  Multiply  U3  by  35. 

OPERATION. 
743 

7  Explanation. — The  factors  of  35   are  5  and  7; 

hence,   multiplying  by  5  and  7,  or  7  and  5,   will 

5201  produce  the  same  result    as   multiplying  by   35; 

5  since  5  X  7  =  35. 


26005 


2.  What  cost  325  bushels  of  potatoes,  at  63  cents  a 
bushel  ?  '  "^ 

3.  What  cost  437  melons,  at  21  cents  a  piece  ? 

4.  What  cost  395  yards  of  muslin,  at  27  cents  a  yard  ? 

5.  What  cost  49  sheep,  at  425  cents  a  head  ? 

6.  What  cost  77  horses,  at  245  dollars  a  piece  ? 

7    What  cost  15  acres  of  land,  at  595  dollars  an  acre  ? 


iO  MULTIPLICATION.  [CHAP.    II 

8.  What  cost  to  bush,  of  wheat,  at  145  cents  a  bushel  ? 

9.  What  cost  18  pounds  of  opium,  at  845  cents  a  pound  ? 

10.  What  cost  21  books  at  95  cents  a  piece  ? 

Art.  43.  MuUiplication  of  abstract  numbers,  when 
there  are  ciphers  on  the  right  of  the  multiplier  or  multi- 
plicand, or  both. 

1.  Multiply  3464  by  2430000. 

OPERATION.  Explanation. — Write  the  numbers  down,  so 

3464  that  the  right  hand  significant  figures  of  the 

3430000  *"^^  factors  shall  come  one  under  the  other ; 

. ^ then  multiply  as  in  Case  2,   Article  40,  and 

10392  bring  the  ciphers  down  on  the  right  of  the 

13856  product. 
10392 


11881520000 


Remark. — This  method  of  operation  is  a  particular  case 
under  Art.  41.  For  in  fact  the  number  3430000,  is  resolved 
into  the  two  factors  343  and  10000.     We  first  multiplied 

the  minuend  by  343,  and  then  the  product  thus  obtained  by  lOOflO,  which  il 

done  by  merely  adding  four  ciphers. 

2.  Multiply  2460000  by  432000 

OPERATION. 

Multiplicand,  2460000 

Multiplier,  432000 


492 
738 
984 


Product,  1062720000000 

3.  Multiply  232  by  10. 

4.  Multiply  682  by  100. 

5.  Multiply  543  by  1000. 

6.  Multiply  4321  by  10000. 

7.  Multiply  3261  by  100000. 

8.  Multiply  246800  by  100000. 

9.  Multiply  2326000  by  43000. 

10.  Multiply  3680200  by  4863000. 

11.  Multiply  12360000  by  43298000. 

12.  Multiply  326000200  by  20046000 


ART.    48.]  MULTIPLICATION  41 

PRACTICAL    QUESTIONS    COMBINING    ADDITION,    SUBTRACTION, 
AND  MULTIPLICATION. 

1.  If  a  wagon  cost  48  dollars,  a  yoke  of  oxen  3  times 
as  much,  lacking  54  dollars,  and  a  span  of  horses  as  much 
as  the  wagon  and  oxen  together;  what  was  the  cost  of  the 
oxen  and  horses  respectively,  and  of  all  ? 

2.  A  man  paid  for  building  his  house  2460  dollars; 
for  his  farm  4  times  as  much,  lacking  986  dollars;  and  for 
his  furniture,  122  dollars  less  than  he  paid  for  building 
his  house.  How  much  did  he  pay  for  all,  and  for  each 
respectively  ? 

3.  Two  persons  start  together  from  the  same  place,  and 
travel  in  the  same  direction.  One  proceeds  at  the  rate  of 
35  miles  a  day;  the  other  at  the  rate  of  42  miles  a  day. 
What  distance  will  they  be  apart  at  the  end  of  45  days  ? 

4.  Bought  19t  acres  of  land,  at  47  dollars  an  acre;  at 
another  time  double  the  number  of  acres,  at  double  the 
price  per  acre,  lacking  12  dollars;  and  at  another  time  as 
many  acres  as  I  had  already  bought,  at  145  dollars  an 
acre.  How  many  acres  did  I  buy,  and  how  much  did  it 
cost  me  ? 

5.  A  farmer  purchased  three  tracts  of  land:  the  first 
contained  195  acres;  the  second  6  times  as  much,  lacking 
203  acres;  the  third  as  much  as  the  first  and  second 
together,  and  45  acres  more.  How  many  acres  did  the 
farmer  purchase,  and  what  did  the  whole  amount  to,  at  45 
dollars  an  acre  ? 

6.  A  planter  sold  465  bales  of  cotton,  at  35  dollars  a 
bale;  and  out  of  the  proceeds  bought  18  mules,  at  65 
dollars  each;  6  span  of  horses,  at  141  dollars  a  span; 
and  4  yoke  of  oxen,  at  95  dollars  a  pair.  How  much 
money  had  he  left  from  the  sale  of  his  cotton  ? 

7.  Mr.  B.'s  yearly  income  is  2890  dollars:  he  pays  for 
house-rent  265  dollars;  his  family  expenses  amount  to  7 
times  as  much,  lacking  199  dollars.  How  much  does  he 
save  annually  ? 

8.  A  man,  having  6894  dollars,  paid  out  of  it  1684 
dollars  for  a  farm;  twice  as  much,  lacking  1999  dollars. 


42  MULTIPLICATION.  [CHAP.    II. 

for  building  a  house;  and  the  remainder,  lacking  989  dol- 
lars, for  farmiug  utensils  and  furnishing  his  house.  What 
was  the  cost  of  the  farm,  the  house,  and  of  the  farming 
utensils  and  furniture  of  the  house,  respectively  ? 

9.  A  has  789  sheep;  B  has  4  times  as  many,  lacking 
999;  and  D  has  45  sheep  more  than  A  and  B  together. 
How  many  sheep  has  B  and  D^respectively,  and  how 
many  have  they  all  ? 

10.  A  is  worth  8967  dollars;  B  is  worth  285  dollars 
more  th^n  A;  and  C  is  worth  as  much  as  A  and  B  to- 
gether, lacking  3794  dollars.  How  much  are  B  and  0 
worth  respectively  ? 

11.  In  an  army  of  8645  men,  1864  men  were  killed  in 
an  action;  and  4  times  as  many  wounded,  lacking  the 
number  that  deserted,  which  was  984.  How  many  men 
were  wounded,  and  how  many  remained  in  the  army  ? 

12.  A  certain  house  is  worth  1460  dollars;  the  farm 
on  which  it  stands  is  worth  5  times  as  much,  +  896  dollars; 
and  the  stock  on  the  farm  is  worth  4  times  as  much  as  the 
house,  lacking  1980  dollars.  What  is  the  value  of  all, 
and  of  the  farm  and  stock  respectively  ? 

13.  A  lends  B  12804  dollars.  B  let  A  have  bank 
stock  to  the  amount  of  2042  dollars;  a  farm  for  5  times 
as  mucli  as  the  bank  stock,  lacking  989  dollars;  and  is  to 
pay  the  remainder  in  cash.  How  much  cash  ought  B  to 
pay  A  ? 

14.  John  has  240  sheep;  James  15  times  as  many  + 
146;  and  Joseph  8  times  as  many  as  both  John  and  James, 
lacking  S999.  Hbw  many  sheep  has  James  and  John, 
and  how  many  have  they  all  t 

15.  If  a  cow  cost  43  dollars;  a  horse  5  times  as  much; 
and  a  farm  9  times  as  much  as  the  cow  and  horse  together, 
lacking  36  dollars;  how  much  more  will  the  farm  cost 
than  5  horses  and  9  cows,  at  the  same  rate  ? 

16.  If  a  quantity  of  sugar  cost  1465  dollars;  a  store 
15  times  as  much,  lacking  9999  dollars;  and  the  lot  on 
which  the  store  stands  2  times  as  much  as  the  sugar  and 
gtore  together,  +  146  dollars;  what  will  be  the  cost  of 
all,  and  of  the  store  and  lot  respectively  ? 


i 


ART.    44.]  DIVISION.  48 

It.  Said  Martha  to  Baldwin,  I  am  worth  245  dollars; 
Baldwin  replies,  that  is  exactly  1  fifth  as  much  as  Ann  is 
worth,  and  1  twelfth  as  much  as  I  am  worth.  How  much 
are  Ann  and  Baldwio  together  worth  ? 

18.  If  a  quantity  of  floar  cost  2864  dollars;  the  store 
in  which  it  is  deposited,  14  times  as  much,  lacking  984 
dollars;  and  the  lot  on  which  the  store  stands  3  times  as 
much  as  the  flour  and  store  together,  +  183  dollars; — 
what  will  be  the  cost  of  all,  and  of  the  store  and  lot  re- 
spectively ? 

19.  A  merchant  bought  12  pieces  of  broadcloth,  each 
piece  containing  32  yards,  at  5  dollars  a  yard  for  th^two 
pieces;  6  dollars  a  yard  for  six  pieces;  and  8  dollars  a 
yard  for  the  remaining  four  pieces.  He  sold  it  all,  at  7 
dollars  a  yard,  did  he  gain  or  lose,  and  how  much  ? 

DIVISION. 

Art.  44.  Division  teaches  how  to  find  the  number 
of  times,  or  part  of  a  time  that  one  number  is  contained 
in  another. 

There  are  three  terms  employed  in  division,  the  Divisor, 
Dividend  and  Quotient.  That  which  is  left,  (if  any,) 
after  the  division,  is  called  the  Remainder; — we  have  not 
called  it  a  distinct  term  of  division,  as  it  is  a  part  of  the 
dividend. 

The  Divisor  is  the  dividing  number.  The  Dividend  is 
the  number  to  be  divided.  The  Quotient  is  the  number 
of  times  the  dividend  contains  the  divisor. 

Division  is  indicated  by  the  symbol,  —.  This  sign 
when  placed  between  two  quantities,  shows  that  the  num- 
ber an  the  left  is  to  be  divided  by  the  one  on  the  right. 
Thus,  8  -f-  4  =  2;  shows  that  8  is  to  be  divided  by  4, 
and  that  the  quotient  is  2.  In  the  division  of  concrete 
numbers,  the  divisor  is  always  considered  abstractly.  The 
quotient  is  a  concrete  number  of  the  same  kind  as  the 
dividend. 

Division  is  also  indicated  by  writing  the  divisor  under 
the  dividend;  thus,  ^^2  =  3. 


44 


SHORT   DIVISION. 


[chap.    II 


DIVISION    TABLE. 


1 
2 

3 
4 
6 
1=  6 


96-r-8= 


S-r- 

6-^ 
9-r- 

12 

15 

18-^ 
21 -j- 

24-^- 

9  27-i- 

:10  30 
:11 

=12  36-^ 


3=  1 
3=  2 
3=  3 
3=  4 
3=  6 
3=  6 
3=  7 
3=  8 
=  9 
3=10 
3=11 
3=12 


4-^4 

8-f-4 
12^4 

16-^4: 

20^-4 

24-f-4: 

28-^4: 
32-J-4: 

364-4: 

40-^4: 
44-f-4: 

48-j-4=12  60-j- 


5-T 

10-^ 

15-^ 

20^ 

25 -f- 

30 

35 

40-4- 

45 

60 

55 


5=  1 
5=  2 
5=  3 
5=  4 
5=  5 
5=  6 
5=  7 
5=  8 
5=  9 
5=10 
5=11 
5=12 


6- 
12- 
18- 
24-H 
30-f- 
36-h 
42-1- 
48-^ 

54-^ 

60-v- 
72-- 


■6=  1 
■6=  2 
■6=  3 


14-h7 
21-^-7 


6=  4  28-i-7= 
6=  635-^7= 
6=  642-h7= 
6=  749-^7= 
6=  856-h7= 
6=  9I63-^7= 
6=10:70-^-7= 
6=ll|77-T-7= 
6=12  84-^7= 


=  11 
=  21 
=  3 
4 
5 
6 
7 


■8= 


9 

18 
27 
36 
45 
54 
63 
72 
81 
90 
99 
108 


^9=  1 
-j-9=  2 
-^9=  3 
-f-9=  4 
-^-9=  5 
H-9=  6 
-j-9=  7 
-j-9=  8 
-h9=  9 
-7-9=10 
-^9=ll 
-J-9=12 


10- 
20- 
30- 
40- 
50- 
60- 
70- 
80- 
90- 

100. 

110. 

120- 


10=  1 
10=  2 
10=  3 
10=  4 
10=  5 
10=  6 
10=  7 
10=  8 
10=  9 
10=10 
10=11 
10=12 


11-r-l 

22-^l 
33-f.l 
44-f-l 
55-^1 
66-^1 
77-^l 

88-r-l 

99-j-l 
110-i-l 
121 -j-1 

132-r-l 


12-hl2= 

24-^12= 

36-^12= 

48-T-12= 

60-^12= 

72-hl2= 

84-^12= 

96-j-12= 

108-r-12= 

120-hl2= 

132-^12= 

144-T-12= 


Short  Division. 

Art.  45.    Division  of   abstract  numbers,  when    the 
divisor  does  not  exceed  12. 
1.  Divide  8245  by  5. 

OPERATION.  Explanation. — Write  the  divisor  at  the 

»  .  left  of  the  dividend,  with  a  curved  Hne  be- 

g  -I  tween  them ;  and  under  the  dividend  draw  a 

horizontal  line.  Begin  at  the  left  and  pro- 
ceed ;  thus,  5  is  contained  in  8  thousands, 
1  thousand  times  and  3  thousands  remain- 
ing. Write  the  1  thousand  down  by  pla- 
cing the  1  under  the  figure  divided.  The 
remainder,  3  thousands,  added  to  2  hun- 
dreds, (which  is  the  same  as  prefixing  the 
3  to  next  figure,)  are  32  hundreds.  5  is  contained  in  32  hun- 
dreds, 6  hundreds  times  and  2  hundreds  remaining.  Write 
the  6  hundreds  down  by  placing  the  6  under  the  last  figure 


Divisor.  Dividend. 
5)  8235 

Quotient,  16  4  7 


ART.    45.]  SHORT  DIVISION.  45 

divided.  The  remainder,  2  hundreds,  added  to  4  tens,  (which 
is  the  same  as  prefixing  the  2  to  the  next  figure,)  are  23  tens. 
5  is  contained  in  23  tens,  4  tens  times  and  4  tens  remaining 
Write  the  4  tens  under  the  last  figure  divided.  The  remain- 
der, 3  tens  added  to  5  units,  are  35  units.  5  is  contained  in 
35  units,  7  units  times,  which  place  under  the  last  figure 
divided ;  and  we  obtain  for  the  quotient,  1647. 

Rr.MARK. — After  the  pupil  thoroughly  understands  the  abore  explantion,  the 
following  may  be  adopted. 

OPERATION.  Explanation. — 5  is   contained  in  8,  1 

Div'sor    D'  *dpnd    ^°^  ^  remaining.     Write  down  the  1  and 

cN   0035      *  prefix  the  remainder  to  the  next  figure. 

^  5  is  contained  in  32,  6  times  and  2  remain- 

Quotient,  1647        '^%.  "^f^  ^^^^  f  ^  ^-  .  ?  ^^  contained 
'  m  23  4  times  and  S  remaining.   5  is  con- 

tained in  35,  7  times  and  no  remainder. 

2.  Divide  1467  by  7. 

OPERATION. 

Divisor.  Dividend 
7)  1  4  6  7  7 


Quotient,        2  0  9  6 — 5  remaider. 

Rf.mark. — The  remainder  6  may  be  divided  by  7,  and  written  with  the  quo- 
tient; thus,  2096f  or  mentioned  simply  as  a  remainder,  as  occasion  requires. 

Proof. — Multiply  the  divisor  by  the  quotient  and  add  in 
the  remainder,  if  there  be  any.  If  this  sum  is  equalto  the 
dividend,  the  work  is  right. 

Remark. — From  what  we  have  already  learned,  we  discover  that  division 
is  the  reverse  of  multiplication,  and  that  either  may  be  used  to  verify,  or 
prove  the  correctness  of  the  work  of  the  other. 

3.  Divide  4682  by  2. 

4.  Divide  3468  by  2. 

5.  Divide  7639  by  3. 

6.  Divide  8472  by  4. 

7.  Divide  89631  by  4. 

8.  Divide  142632  by  '^ 

9.  Divide  34682  by  6 

10.  Divide  24673  by  5. 

11.  Divide  147268  by  5. 


46  SHORT   DIVISION.  ,      [CHAP.    II. 

12.  Divide  4*76846  by  9. 

13.  Divide  4t6342  by  8. 

14.  Divide  8462324  by  8. 

15.  Divide  8496T23  by  9. 

16.  Divide  846t232  by  8. 
11.  Divide  246832  by  10. 

18.  Divide  46t232  by  11. 

19.  Divide  2468324  by  12. 


PRACTICAL     QUESTIONS. 

1.  If  9  acres  of  land  cost  2250  dollars,  what  will  1 
acre  cost  ? 

2.  If  8  horses  cost  1696  dollars,  what  will  1  horse 
cost  ? 

3.  If  a  man  travel  693  miles  in  9  days,  how  far  does  he 
travel  in  1  day  ? 

4.  Divide  1648  acres  of  land  equally  among  8  indivi- 
duals. 

5.  If  6  horses  sell  for  1332  dollars,  what  will  be  the 
average  sum  received  for  each  ? 

6.  A  man  bought  12  tons  of  hay  for  192  dollars;  how 
much  did  he  pay  a  ton  ? 

1.  A  boy  sold  11  rabbits  for  286  cents;  how  much  did 
lie  receive  a  piece  ? 

8.  A  girl  spent  342  cents  for  oranges,  at  3  cents  a 
piece;  how  many  oranges  did  she  buy  ? 

9.  Divide  68425  dollars  equally  among  t  sons. 

10.  How  many  barrels  of  flour,  at  6  dollars  a  barrel, 
can  be  bought  for  25218  dollars  ? 

11.  At  8  dollars  a  cord,  how  many  cords  of  wood  can 
be  bought  for  1928  dollars  ? 

12.  At  5  dollars  a  barrel,  how  many  barrels  of  cider 
can  be  bought  for  1465  dollars  ? 

13.  If  in  1  week  there  are  Y  day,  how  many  weeks  are 
there  in  365  days,  (one  year)  ? 

14.  A  man  bought  a  store  for  3*192  dollars,  which  was 
3  times  as  much  as  his  house  cost  him;  how  much  did  his 
house  cost  him  ? 


ART.  46."]  LONG   DIVISION.  41 

15.  A  drover  bought  12  oxen  for  It 64  dollars;  how 
much  was  the  average  cost  of  each  ? 

16.  A  laborer  worked  12  months  for  288  dollars;  how 
much  did  he  receive  a  month  ? 

IT.  A  is  worth  15T95  dollars,  which  is  5  times  as  much 
as  B  is  worth,  and  B  is  worth  3  times  as  much  as  C;  how 
much  are  B  and  C  worth  respectively  ? 

18.  A's  house  cost  2358  dollars,  which  is  3  times  as 
much  as  the  furniture  of  the  house  cost;  what  was  the 
cost  of  the  furniture  ? 

19.  Says  A  to  B,  I  have  T4  sheep;'  B  replies,  that  is 
just  1  tenth  of  my  number,  which  is  4  times  C's  number; 
how  many  sheep  has  C  ? 

20.  Edward  is  worth  2000  dollars,  which  is  3  times 
Luther's  fortune,  lacking  TOO  dollars:  and  Caleb  is  worth 
4  times  as  much  as  Edward  and  Luther  together +400 
dollars.     What  is  the  fortune  of  each  ? 


Long  Division. 

Art.  46.  Division  of  abstract  numbers  in  general. 
1.  Divide  4379  by  24. 

OPERATION.  Explanation. — Write  the  divi- 

sor on  the  left  of  the  dividend; 
and  the  quotient  on  the  right, 
separating  them  with  a  curved 
line,  and  proceed  thus:  24  is 
Divisor.  Dividend.  Quotient,  contained  in  43  hundreds  and  79, 
24)      4  3  7  9(100  }  hundred  times  ;  write  the  100 

2400  80         ^"         quotient,  100  times  24  is 

2  2400,    which    l3eing    subtracted 

I  Q  Y  Q      from  the  dividend,  leaves  1979, 

10  90       18911     24  is  contained  in  1979,  80  times; 

^^^^'        -^  °  -  2T     ^rite  the  80  in  the  quotient,  80 

times  24  are  19*20,  which  being 

subtracted  from  the  1979,  leaves 

59.     24  is    contained  in    59,  2 


S3  "^     M     " 

C    C   C-- 


59 

48 


Remainder,     1  1  *!«^f '  ™*^  ^^^Z  '""a^^  2"t 

'  tient.     2  tunes  24  are  48,  which 


48  LONG   DIVISION.  [cHAP.    H 

being  subtracted  from  the  59,  leaves  11.     Dividing  the  11  by 
24  we  have  |^,  which  annex  to  the  quotient. 

Remark. — After  the  pupil  comprehends  the  above  explanation,  the  follow- 
ing may  be  adopted. 

OPERATION.  Explanation. — 24  is  contain- 

Divisor.  Dividend.  Quotient,     ©d  in  43,  1  time.    Write  the  1  in 

24)         4379         (182^^       *h®  quotient.     1  times  24  is  24, 

24  ^         which  being  subtracted  from  43, 

leaves  19.     Bring  down  the  next 

197  figure  of  the  dividend.      24  is 

192  contained  in  197,  8  times ;  write 

the  8  in  the  quotient.     8  times 

59  24  are   192,  which  being  sub- 

48  tracted  from  197,  leaves  5.  Bring 

down  the  next  figure  of  the  divi- 

11  dend.     24  is  contained  in  59,  2 

times.  Write  the  2  in  the  quo- 
tient, 2  times  24  are  48,  which  being  subtracted  from  59,  leaves 
11.  Divide  the  remainder  by  24 ;  thus,  ^^ ;  and  place  it  in  the 
quotient. 

Proof  hy  the  excess  of  9'^. — Find  the  excess  of  9's  in  the 
divisor  and  quotient  respectively,  and  also,  the  excess  of 
9's  in  the  product  of  these  two  excesses;  and  if  this  last 
excess  is  equal  to  the  excess  of  9's  in  the  difference  be- 
tween the  dividend  and  remainder,  the  work  is  right. 

Take  for  illustration  the  above  example  : 

Divisor,  24  =6  excess. 

Quotient,      182  =  2  excess. 

Dividend,    4379 
Remainder,     11 

Difference,  4368  =  3  J  excess, 

2.  Divide  4368  by  13. 

3.  Divide  369a  by  15. 

4.  Divide  8041  by  11. 

5.  Divide  5490  by  15. 

6.  Divide  1242  by  2t. 
1.  Divide  66384  by  24. 


LET.  46.]  LONG   DIVISION.  49 

8.  Divide  108220  by  28. 

9.  Divide  18336  by  24. 

10.  Divide  2841  by  29. 

11.  Divide  3570  by  15. 

12.  Divide  6048  by  72. 

13.  Divide  3607344  by  24. 

14.  Divide  949073  by  73. 

15.  Divide  9334949  by  307. 

16.  Divide  789591  by  213. 

17.  Divide  86431  by  342. 

18.  Divide  986321  by  412. 

19.  Divide  2364  by  82. 

20.  Divide  146832  by  147. 

21.  Divide  246832  by  432. 

22.  Divide  846324  by  1432. 

23.  Divide  98476324  by  1463. 

24.  Divide  1476324  by  1482. 

25.  Divide  47632463  by  24801. 

26.  Divide  476784631  by  1472 

27.  Divide  48468234  by  423. 

28.  Divide  123456789  by  846. 

29.  Divide  987654321  by  146. 

30.  Divide  987644698321  by  3223. 

PRACTICAL  QUESTIONS. 

1.  If  one  man  can  accomplish  a  certain  piece  of  work 
in  494  days,  bow  many  days  will  it  take  38  men  to  do  the 
same  ? 

2.  If  99  sheep  cost  396  dollars,  what  will  1  sheep  cost  ? 

3.  If  97  acres  of  land  cost  22989  dollars,  how  much  is 
that  an  acre  ? 

4.  What  cost  1  barrel  of  flour,  if  36  barrels  cost  288 
dollars  ? 

5.  If  in  89  books  there  are  28035  pages,  how  many 
pages  on  an  average  in  a  book  ? 

6.  If  an  iceberg  move  at  the  rate  of  25  miles  a  day, 
how  many  days  would  it  be  in  moving  from  the  north  pole 
to  the  equator,  it  being  about  6250  miles  ^ 

8 


60  LONG   DIVISION. 


CHAP.  II. 


t.  If  a  horse  can  travel  54  miles  in  a  day,  how  many 
days  will  it  take  it  to  travel  10854  miles  ? 

8.  If  15  months'  wages  amount  to  525  dollars,  how 
much  is  that  a  month  ? 

9.  If  Ml  quails  are  sold  for  128*7  cents,  how  much  is 
that  a  piece  ? 

10.  If  38  baskets  of  peaches  are  sold  for  2850  cents, 
how  much  is  that  a  basket  ? 

11.  A  drover  bought  cattle,  at  3t  dollars  a-head,  and 
paid  for  them  8732  dollars,  how  many  did  he  buy  ? 

12.  How  many  barrels  of  molasses,  at  It  dollars  a 
barrel,  can  be  bought  for  3604  dollars  ? 

13.  How  many  pieces  of  cloth,  at  95  dollars  a  piece,  can 
be  bought  for  3385  dollars  ? 

14.  If  63  gallons  make  1  hogshead,  how  many  hogsheads 
will  1449  gallons  make  ? 

15.  For  1016  dollars,  how  many  yards  of  broadcloth 
can  be  bought,  at  8  dollars  a  yard  ? 

16.  If  a  steamship  can  cross  the  Atlantic  Ocean,  a  dis- 
tance of  3000  miles,  in  9  days;  how  many  miles  does  the 
ship  go  daily  ? 

17.  Baldwin's  income  is  2555  dollars  a  year,  how  much 
is  that  a  day,  allowing  the  year  to  consist  of  365  days  ? 

18.  Walter  purchased  a  farm  containing  235  acres,  for 
4230  dollars;  how  many  dollars  did  he  pay  an  acre  ? 

19.  In  how  many  days  could  27  men  accomplish  the 
same  amount  of  work,  that  1  man  could  in  594  days  ? 

20.  If  a  railroad  car  move  at  the  rat*^  of  625  miles 
a  day,  in  how  many  days  would  it  go  around  the  earth, 
the  distance  being  about  25000  miles  ? 

Art.  47.  To  divide  one  number  by  another,  when  the 
divisor  is  a  composite  number. 

Resolve  the  number  into  two  or  more  fado's,  then  divide  by 
one  of  these  factors,  and  the  quotient  thus  obtained  by  another 
factor, — proceed  in  the  same  way  till  all  the  factors  have  be 
come,  divisors,  and  the  last  quotient  obtained  will  be  the 
answer  required. 

Art.  48.  To  find  the  true  remainder. 


ART.    48.]  LONG  DITISION.  51 

To  the  sum  of  the  products  of  each  remoXTider  into  all  the 
divisors  preceding  the  one  that  produced  it,  add  the  first  re- 
mainder, and  this  sum  will  be  the  true  remainder. 

1.  Divide  2486  by  105. 

The  factors  of  105  are  3,  5  and  t. 

OPERATION. 

1  3)2486 

2.  5)828—2  Ist  remainder. 

„  3.  7)165—3  2nd 

Quotient,        23—4  3rd        " 

Explanation — The  small  figures  1,  2,  3,  on  the  left  of  the 
divisors  are  used  to  designate  the  numbers  that  have  become 
dividends.  The  ^d  remainder  is  the  same  as  the  3d  dividend, 
but  a  unit  of  the  Zd  dividend  is  equal  to  5  units  of  the  2nd 
dividend ;  and  a  unit  of  the  2nd  dividend  is  equal  to  3  units  of 
the  \st  dividend,  since  the  1st  and  2nd  dividends  have  been 
divided  respectively  by  3  and  5.  Therefore,  a  unit  of  the  3d 
remainder  is  equal  to  5x3  units  of  the  \st  remainder,  and  4 
units  of  the  3d  remainder  is  5x3x4=60  units  of  the  first  re- 
mainder. For  a  similar  reason  3  units  of  the  2nd  remainder  ia 
equal  to  3x3=9  units  of  the  Ist  remainder.  To  the  sum  of 
these  products  add  the  first  remainder,  and  we  have  60-f-9-j-2=71 
the  true  remainder. 

2.  Divide  4898  by  21. 

3.  Divide  9042  by  15. 

4.  Divide  11128  by  1155. 

5.  A  man  bought  15  horses  for  2910  dollars;  how 
much  was  that  a  piece  ? 

6.  If  2T  barrels  of  flour  cost  250  dollars,  how  much  is 
that  a  barrel  ? 

7.  A  wealthy  merchant  distributed  588  yards  of  cloth 
equally  among  49  poor  individuals;  how  many  yards  did 
they  receive  a  piece  ? 

8.  A  drover  paid  1456  dollars  for  cattle,  giving  56 
doi]ars  a  head;  how  many  cattle  did  he  buy  ? 

9.  A  farmer  bought  98  acres  of  land  for  2178  dollars; 
bow  much  did  he  pay  an  acre  ? 


52 


LONG   DIVISION. 


.  [chap.   II. 


10.  In  a  certain  corn-field  there  are  5229  hills  of  corn, 
and  63  rows;  how  many  hills  in  a  row  ? 

Art.  49.  Division  of  abstract  numbers,  when  the  divi- 
sor, or  dividend,  or  both  have  ciphers  on  the  right. 
1.  Divide  82468524  by  24500. 

OPERATION. 


3. 

4. 

5. 

6. 

•7. 

8. 

9. 
10. 
11. 
12. 
13. 
14. 
15. 


245!00)824685|24(33662V^ 
735 

896 
-      735 

1618 
1470 

1485 
1470 


Rkmark.— I  cu|  off  the 
ciphers  on  the  rigfct  of  the 
divisor,  and  a,s  many  places 
on  the  right'^jf  the  divi- 
dend.  Alter  the  division, 
affixing  the  remainder  to 
the  quotient  with  the  divi- 
sor  under  it,  and  a  hori- 
zontal line  between  them, 
and  we  have  no  remainder. 


Remainder,  1524 


Div 
Div 
Div 
Div 
Div 
Div 
Div 
Div 
Div 
Div 
Div 
Div 
Div 
Div 


de  2468  by  10. 

de  374232  by  100. 

de  468324  by  1000. 

de  36842  by  1100. 

de  468234  by  450. 

de  476324  by  4810. 

de  846324  by  7800. 

de  14786324  by  48300. 

de  246832  by  470. 

de  2476800  by  470. 

de  8468300200  by  47600. 

de  12468300200  by  3680. 

de  4780024680000  by  8496000. 

de  8468476008470000  by  84000. 


ABSTRACT    EXAMPLES  IN  THE    FUNDAMENTAL  RULES. 

Art.  50.  Quantities  enclosed  in  a  parenthesis,  some- 
times called  the  sign  of  aggregation,  (  ),  are  to  be  subject- 
ed to  the  same  operatien     Thug,  (3+6 — 2)x5,  denotes 


ART.  50.]  LONG   DIVISION.  58 

tHat  the  sum  of  3  and  6,  lacking  2  is  to  be  multiplied  by 
5,  the  product  of  which  is  35. 

1.  What  is  the  value  of  the  expression,  (465 — 2*1+14:0) 
X8.? 

2.  What  is  the  value  of  the  expression,  .(846+41  +  96) 
X25? 

3.  What  is  the  value  of  the  expression,  (891— 4t+86) 
XI— 184.? 

4.  What  is  the  value  of  the  expression,  464+ (843— 
81  +  9)  X416— 461.? 

5.  What  is  the  value  of  the  expression,  461 — 189  + 
(88— 14  +  215)X91? 

6.  What  is  the  value  of  the  expression,  (462+1—146) 
X  (84— 14  +  115).? 

1.  What  is  the  value  of  the  expression,  96+ (144 — 91) 
X(86— 41— 189)-7-93? 

8.  What  is  the  value  of  the  expression,  (41 — 23+12) 
-7-9  +  (98+4)X(144— 91)? 

9.  What  is  the  value  of  the  expression,  (14  +  1)X2  + 
(256—25)^21? 

•  10.  What  is  the  value  of  the  expression,  (8 9 6 — 116) -r 
144  +  (214  +  82)X(86— 41  +  8)? 

PRACTICAL   QUESTIONS    COMPRISING    THE     FOUR    FUNDAMENTAL 
RULES. 

1.  Henry  has  4684  dollars,  which  lacks  248  dollars  of 
being  4  times  James'  fortune;  and  Jackson  is  worth  3 
times  as  much  as  Henry  and  James  together,  lacking  3421 
dollars.  How  much  money  have  James  and  Jackson 
respectively  ? 

2.  A  man  bought  an  equal  number  of  cows  and  horses 
for  9120  dollars;  for  the  cows  he  gave  234ollars  a  piece  ; 
and  for  the  horses  91  dollars  a  piece;  how  many  of  each 
did  he  buy .? 

3.  A  merchant  expended  336  dollars  for  an  equal  num- 
ber of  yards  of  broadcloth,  consisting  of  three  different 
kinds;  the  first,  at  5  dollars  a  yard;  the  second,  at  1  dol- 


54  LONG   DIVISION.  [CHAP.  II. 

lars ;  and  the  third,  at  9  dollars  a  yard.     How  many  yards 
of  each  kind  did  he  buy  ? 

4.  A  farmer  sold  an  equal  number  of  chickens,  ducks, 
and  geese  for  3540  cents;  the  chickens,  at  12  cents  each; 
the  ducks,  at  37  cents  each;  and  the  geese,  at  69  cents 
each.     How  many  of  each  kind  did  he  sell  ? 

5.  A  gave  1  eighth  of  89648  dollars  for  a  farm,  which 
was  237  dollars  more  than  it  was  worth;  how  much  was 
the  farm  worth  ? 

6.  Light  moves  about  11550000  miles  a  minute ;  at  this 
rate^  how  long  would  light  be  in  passing  from  the  sun  to 
the  earth,  a  distance  of  95000000  of  miles  ? 

7.  The  product  of  two  numbers  is  91096;  and  one  of 
the  numbers  is  472.     What  is  the  other  number  ? 

8.  The  quotient  arising  from  dividing  one  number  by 
another  is  345:  the  dividend  is  273585.  What  is  the  divi- 
sor ? 

9.  The  quotient  arising  from  a  certain  division  is  437; 
the  divisor  is  413;  and  the  remainder  247.  What  is  the 
dividend  ? 

10.  A  farmer's  yearly  income  was  19437  dollars.  He 
paid  for  repairing  his  house  313  dollars;  for  hired  help  on 
his  farm,  5  times  as  much,  lacking  65  dollars;  and  for 
traveling  expenses  2463  dollars.  How  much  does  he  save 
yearly  ? 

11.  Bought  45  barrels  of  flour  for  225  dollars;  for  what 
must  it  be  sold  a  barrel  to  gain  135  dollars,  and  what 
will  be  the  gain  on  each  barrel  ? 

12.  Bought  130  acres  of  land  for  5850  dollars;  and  sold 
112  acres  of  it,  at  75  dollars  an  acre,  and  the  remainder 
for  what  it  cost;  how  much  did  I  gain  by  the  bargain  ? 

13.  Bought  150  acres  of  land  for  9750  dollars;  and 
sold  apart  of  it  for  7140  dollars,  at  85  dollars  an  acre; — 
how  many  acres  had  I  remaining,  and  how  much  did  I 
gain  on  every  acre  sold  ? 

14.  A  farmer  sold  corn  for  864  dollars;  wheat  for  895 
dollars;  rye  and  oats  for  3  times  as  much  as  he  received 
for  the  corn  and  wheat  together,  lacking  148  dollars.  Out 
of  these  proceeds  he  bought  6  span  of  horses,  at  275  dol- 


ART.    60.]  LONG    DIVISION.  65 

lars  ^  span;  5  yoke  of  oxen,  at  125  a  pair;  and  the  re- 
mainder, lackiug  738  dollars,  he  paid  for  land,  at  65  dol- 
lars an  acre.     How  many  acres  did  he  buy  ? 

15.  Bought  195  acres  of  land,  at  84  dollars  an  acre, 
which  cost  12  times  as  much  as  I  paid  for  a  span  of  fine 
horses.  I  have  noiw  1468  dollars  remaining.  JIow  much 
money  had  I  at  first  ? 

16.  If  an  army  of  6000  men  have  provisions  for  5 
months,  and  4400  men  be  disengaged;  how  long  will  the 
same  provisions  serve  the  remainder  ? 

n.  A  certain  tradesman  can  earn  54  dollars  a  month,  but 
his  necessary  expenditures  are  29  dollars  a  month.  He  de- 
sires to  purchase  a  farm  containing  75  acres,  worth  35  dol- 
lars an  acre.  In  what  tune  can  he  save  money  enough  to 
make  the  purchase  ? 

18.  Sold  to  my  neighbor  12  cords  of  wood,  at  5  dollars 
a  cord;  65  barrels  of  corn,  at  2  dollars  a  barrel;  45  head 
of  cattle,  at  65  dollars  a  head.  In  payment,  I  take  5  sacks 
of  coffee,  at  15  dollars  a  sack;  25  barrels  of  sugar,  at  15 
dollars  a  barrel;  2405  dollars  in  cash;  and  the  remainder, 
in  molasses,  at  26  dollars  a  barrel.  How  many  barrels  of 
molasses  ought  I  to  receive  ? 

19.  A  drover  bought  a  certain  number  of  cattle  for 
8050  dollars,  and  sold  a  certain  number  of  them  for  6231 
dollars,  at  63  dollars  each,  and  gained  on  those  he  sold 
1683  dollars;  how  many  did  he  buy  at  first,  and  how 
much  did  he  gain  a  piece  on  those  he  sold  ? 

20.  A  speculator  gave  18810  dollars  for  a  certain  num- 
ber of  acres  of  land,  and  sold  a  part  of  it  for  1990  dollars, 
at  85  dollars  an  acre,  and  by  so  doing,  lost  10  dollars  on 
each  acre;  for  how  much  must  he  sell  the  remainder  an 
acre  to  gain  2180  dollars  by  the  operation  ? 

21.  A  farmer  gave  37620  dollars  for  a  farm,  and  sold  a 
certain  number  of  acres  of  it  for  15980  dollars,  at  85  dol- 
lars an  acre,  and  by  sodding  lost  20  dollars  an  acre;  for 
how  much  must  he  sell  the  remainder  an  acre  to  gain  4360 
dollars  by  the  operation  ? 


66  DENOMINATE   NUMBERS.  [CHAP.  III. 


CHAPTER  III. 

Tables  of  Money,  Weights  and  Measures. — Addition, 
Subtraction,  Multiplication,  and  Division  of  Polyno- 
mials, OR  Denominate  Numbers. 

Art.  5 1  •  a  Simple  Number  is  either  a  unit,  or  a  col- 
lection of  units  considered  abstractly,  that  is,  without 
reference  to  any  particular  thing;  as  8,  16,  24,  &c. 

Art.  52,  A  Concrete,  or  Denominate  Number,  is 
either  a  unit,  or  a  collection  of  units  having  reference  to 
some  particular  thing;  as  4  feet,  5  dollars,  8  hours,  25  men, 
&c.  The  measuring  unit  of  any  quantity  is  a  similar  con- 
crete unit,  by  means  of  which  the  quantity  is  expressed 
numerically. 

Art.  53.  A  Monomial  in  Algebra,  is  a  quantity  of  one 
term  only;  it  may  also,  with  propriety,  be  applied  to  an 
Arithmetical  number,  when  it  is  expressed  by  a  single  name 
of  a  measuring  unit;  as,  5  dollars,  1  bushels,  10  men,  &c. 

Art.  54.  A  Polynomial  in  Algebra  is  a  quantity  con- 
sisting of  many  terms;  it  may  also  be  applied  to  denominate 
numbers,  signifying  a  quantity  of  many  names;  as,  2  cwt. 
3  qrs.  15  lbs.,  &c.  It  is,  however,  more  generally  applied 
to  an  abstract  number  consisting  of  many  terms ;  as, 
(4  +  6  +  8  -I-  a,)  &c. 

Table  of  United  States  Currency. 


Mills 

make     1  Cent, 

marked  c. 

Cents 

"        1  Dime, 

"       d. 

Dimes 

"        1  Dollar, 

"       $. 

Dollars 

"        1  Eagle, 

"      E. 

Art.  55.  It  will  be  observed  that  the  measftl-ing  units 
in  the  Dnit*ed  States  currency  increase  in  a  tenfold  ratio, 
as  in  abstract  numbers.  Hence  this  currency  will  be 
treated  of  under  Decimal  Fractions.  The  measuring  units 
of  other  kinds  of  quantity,  increase  from  lower  te  higher 


ART.  68.]  ATOIRDUPOIS    WEIGHT.  5t 

orders  according  to  the  scales  of  increase  given  in  the  fol- 
lowing tables. 

English  or  Sterling  Money. 
Art.  56.  English  Money  is  the  currency  of  England. 
Its  denominations  are  Pounds,  Shillings,  Pence,  and  Far- 
things. 

TABLE. 

4  Farthings  (far.  or  qr.)  make  1  Penny,  marked  d. 
12  Pence  "  1  Shilling,         "  s. 

20  ShilUngs  "  1  Pound,  "  £. 

Troy  Weight. 

Art.  57.  By  this  weight  are  weighed  gold,  silver,  and 
jewels. 

Remark. — The  original  of  all  weights  used  in  England,  was  a  grain  of 
wheat,  taken  from  the  middle  of  the  ear  ;  32  of  these,  dried,  were  to  make 
1  jiennyweight. 

Since  then  it  was  agreed  to  divide  the  same  pennyweight  into  24  equal 
parts,  still  called  grains,  being  the  least  weight  in  common  use. 

TABLE. 

24  Grains  (gr.)  make     1  Pennyweight,     marked,    pwt. 

20  Pennyweights  "         1  Ounce,  "  oz. 

12  Ounces  .  "        1  Pound,  "         lb. 


Avoirdupois  Weight. 

Art.  58.  Avoirdupois  Weight  is  used  to  weigh  all 
things  of  a  course  nature,  as  groceries,  some  liquids,  and 
all  metals,  except  gold  and  silver. 

table. 

16  Drams  (dr.)  make   1  Ounce,                    marked  oz. 

16  Ounces             -  "1  Pound,                           "       lb. 

25  Pounds*  "       1  Quarter,                        "       qr. 

4  Quarters  "       1  Hundred  Weight,        "       cwt. 

20  Hundredweight  "      1  Ton,                              "      T. 

•  Note. — In  buying  and  selling  articles,  it  is  customary  to  call  25  pounds,  I 
qr.,  instead  of  28}  and  100  pounds,  1  cwt.,  instead  of  112  pounds,  as  was  for 
xneily  done. 

3* 


68 


DENOMINATE    NUMBERS. 


[chap.  III. 


Apothecaries'  Weight. 

Art.  59.  Apothecaries'  Weight  is  used  in  compound- 
ing, or  weighing  small  quantities  of  medicines,  as  for  pre- 
scriptions. But  medicines  and  drugs  by  the  quantity,  are 
generally  bought  and  sold  by  avoirdupois  weight.  The 
pound  and  ounce  Apothecaries'  Weight  equals  the  pound 
and  ounce  Troy  Weight. 


20  Grains  (gr.) 
3  Scruples 
8  Drams 

12  Ounces 


make 


1  Scruple, 
1  Dram, 
1  Ounce, 
1  Pound, 


marked 


Cloth  Measure. 
Art.  60.  Cloth  Measure  is  used  in  measuring  cloth, 
lace,  ribbons,  and  all  other  articles  sold  by  the  yard. 


21  Inches  (in.) 
4  Nails,  or  9  in. 

4  Quarters 
3  Quarters 

5  Quarters 

6  Quarters 


TABLE. 

make  1  Nail,  marked  na. 

"       1  Quarter  of  a  yard,        "  qr. 

"       1  Yard,  "  yd. 

"       1  Ell  Flemish,  "  E.  FL 

"       1  Ell  English,  "  E.  E. 

"       1  Ell  French,  "  E.  Fr. 


Long  Measure. 
Art.  61.  This  measure  is  used  in  measuring  distances. 


12  Inches  (in.) 

3  Feet 
6\  Yards,  or  16^  feet, 
40  Rods 

8  Furlongs 

3  Miles 
60  Geographic  miles, 

or    69|    statute    or 

league  miles 
360  Degrees 


table. 

lalj 

u 
u 

C( 

u 

e  1  Foot, 
1  Yard, 
1  Rod,  Pol 
1  Furlong, 
1  Mile, 
1  League, 

1  Degree, 

1  Circle, 

marked 


ft. 

rd. 
fur 

m 
lea 


deg.  or 


ART.    62.] 

4  Inches 
6  Feet 


SUPERFICIAL,    OR    SQUARE    MEASURE. 

make  1  Hand. 


59 


Used  in  measuring 
the  height  of  horses, 

1  Fathum,  sy^^i^^r'''''''"^ 

'I  depths  at  sea. 


Superficial,  or  Square  Measure. 

Art.  62.  This  measure  is  used  for  measuring  all  kinds 
of  surfaces,  such  as  land,  boards,  plastering,  and  everything 
else,  in  which  length  and  breadiU  only  are  considered. 

4  feet.  A    Square  is   a  figure 

havirip;  four  equal  sides, 
and  four  equal  angles,  or 
four  rigi\b  angles. 

This  diagram  is  called 
four  feet  square^  as  it  is 
four  feet  each  way.  Each 
of  the  small  squares, 
(within  the  large  square,) 
represents  1  square  foot. 
There  are  4  square  feet 
in  each  row,  and  4  rows 
in  the  whole  square; 
therefore,  there  are  4 
times  4  square  feet,  equal 
to  16  square  feet^  in  4  feet  square ;  hence  there  is  a  difference 
of  12  square  feet  between  4  feet  square,  and  4  square  feet.  The 
4  square  feet  is  represented  by  the  squares  1,  2,  3,  and  4 ;  and 
the  4  feet  square,  by  the  large  square  which  contains  16  square 
feet.  From  the  above  we  infer  that  the  superficial  contents  of 
a  square,  or  any  rectangular  figure  is  found  by  multiplying  its 
length  with  its  width. 


1 

Square 
foot. 

2 

3 

4 

I 

4  feet. 


TABLE. 


144  Square  Inche8(sq.in.)  make  1  Square  Foot,  marked  sq.  ft. 
9  Square  Feet  "     1  Square  Yard,  "     sq.  yd. 


M. 


30]-  Square  Yardu 

"     1  Sq.  Rod,  or  Pole,  " 

P. 

40  Square  Rodb  cr  Poles 

"    1  Rood, 

R. 

4  Roods 

"     1  Acre,                      " 

A. 

640  Acres 

"    1  Square  mile,          " 

S. 

60 


DENOMINATE    NUMBERS. 


[chap.  hi. 


Surveyor's  Measure. 
In  measuring  land,  roads,  &c.,  Gunter's  chain  is  used; 
the  length  of  which  is  4  rods,  or  66  feet. 


TABLE. 


7yVo  Inches  (in.)                  make  1  Link 

marked 

li. 

25  Links                                 "     1  Rod,  or  Pole, 

u 

p. 

4  Poles,  or  100  links            "     1  Chain, 

(C 

cha. 

10  Chains                                 "     1  Furlong. 

u 

fur 

8  Furlongs,  or  80  chains,    "     1  Mile, 

c< 

M. 

10  Square  Chains                   "     1  Acre, 

li 

A. 

Solid,  or  Cubic  Measure. 

Art.  63.  This  measure  is  used  in  measuripg  all  things 
that  have  length,  breadth,  and  thickness;  as  timber, 
boxes  of  goods,  capacity  of  ships,  &c.,  &c. 

A  cube  is  a  solid,  bounded  by  six  equal  and  square 
sides.  ^ 

If  each  of  the  sides  of  a  cube  is  1  foot  it  is  called  a 
cubic  foot.  If  each  of  the  sides  of  a  cube  be  3  feet  =  1 
yard,  it  is  called  a  cubic  yard. 

The  annexed  diagram  represents 
a  cubic  yard.      Since  each  of  the 
sides  of  a  cubic  yard  is  3  feet  each 
way ;  each  of  these  sides  will  con- 
tain  9  square  feet.      If  from  one  j^I 
side  of  this  cube  we  cut  off  a  piece  l] 
1  foot  in    thickness,   we  evidently  % 
have  9  solid  feet ;  and  as  the  whole  ^ 
block  is  3  feet  thick,  it  must  con- 
tain  3    times   9   =   27   solid  feet. 
Hence,  to  find   the  solid  contents 
of  a  cube,  we  multiply  its  length,  breadth^  and  thickness  together. 


3  feet=l  yd. 


TABLE. 

1728  Cubic  inches  (cu.  in.)  make  1  Cubic  foot,  marked  cu.  ft- 
27     "    feet  "      1       "    yard,      "       cu.  y<J 

40     «'    feet  "      1  Ton,  "       T. 

16     "    feet  "      1  Cord  foot,         "       c.  f^ 

8  Cord  feet,  o- 
128  Cubic  feet 


1  Cord  of  wood. 


C. 


ART.  66.] 


DRY   MEASURE. 


\ 

61 


Wine  Measure. 
By  this  measure  all  liquids,  except  beer  ar^ 


Art.  64 
measured. 

The  wine  gallon  contains  231  cubic  inches. 

TABLE. 


4     Gills  (gi.)                make     1  Pint, 

marked  pt. 

2     Pints 

1  Qnart, 

"         qt. 

4     Quarts                          ' 

1  Gallon, 

gal. 

31J  Gallons 

1  Barrel, 

"         bar. 

42"  Gallons                         ♦ 

'        1  Tierce, 

"         tier. 

63    Gallons                          * 

'        1  Hogshead, 

hhd. 

2   Hogsheads                   ' 

'        1  Pipe, 

«         pi. 

2   Pipes 

'        1  Tun, 

"        tun. 

Ale,  or  Beer  Measure. 

Art.  65.  By  this  measure  ale,   beer,    and   milk,    are 
measured. 

The  beer  gallon  contains  282  cubic  inches. 


table. 


2 

Pints  (pt.)                   make     1  Quart,        marked  qt. 

4 

Quarts                              "         1  Gallon,             "         gal. 

6 

Gallons                             "         1  Barrel,             "        bar. 

1| 

Barrels,  or  54  Gallons    "        1  Hogshead,       "        hhd. 

Dry  Measure. 

Art.  66.  Grain,  salt,  coal,  &c.,  are  measured  by  Dry 
Measure. 

The  Dry  Gallon  contains  268f  cubic  inches.  The  Win- 
chester Bushel  contains  2150f  i  cubic  inches.  A  Cylindri- 
cal Measure,  8  inches  deep  and  18^  inches  in  diameter, 
contains  1  bushel. 


TABLE. 


2  Pints  (pt.) 

make     1  Quart,               marked 

qt. 

8  Quarts 

«      1  Peck,                     « 

t. 

4  Pecks 

"       1  Bushel,                   " 

36  Bushels 

"       1  Chaldron               " 

ch. 

32  Bushels 

"      1  Chaldron  in  the  United  t 

5tate& 

62 


DENOMINATE    NUMBERS. 


[chap.  IIL 


Circular  Measure. 

Art.  67.  This  measure  is  used  in  noting  any  part  of 
the  circumference  of  a  circle;  it  is  also  used  in  reckoning 
latitude  and  longitude^  and  the  revolutions  of  the  heavenly 
bodies. 


60  Seconds  (") 

60  Minutes 

30  Degrees 

12  Signs,  or  360  Degrees 


TABLE. 

make 


1  Minute, 
1  Degree, 
1  Sign, 
1  Circle, 


marked 


Measure  of  Time. 

Art.  68.  This  measure  is  applied  to  the  divisions  and 
sibdivisions  of  time. 


60   Seconds  (sec.) 

make 

1  Minute,  marked  min. 

60   Minutes 

a 

1  Hour. 

hr. 

24   Hours               » 

(I 

1  Day, 

da. 

7   Days 

a 

1  Week, 

''        wk. 

4   Weeks 

ii 

1  Month, 

"        mo. 

12   Calendar  months,  or 

52  Weeks,  1  day,  and  6  hours 

1  Year, 

"        yr. 

)65   Days,  6  hours,  (nearly,) 

u 

1  Year, 

"      yr. 

The  Solar  year  consists  of  365  days,  5  hours,  48  min- 
utes, and  5 If  seconds,  and  is  the  exact  time  in  which  the 
Earth  performs  one  revolution  around  the  Sun. 

The  Civil  year  consists  of  365  days.  Hence,  the  dif- 
ference between  the  Solar  and  the  Civil  year  is  nearly  6 
hours,  making  about  1  day  in  4  years. 

As  the  difference  between  the  Solar  and  the  Civil  year 
confused  dates,  Julius  Caesar  made  the  first  correction  of 
the  calender  by  introducing  an  intercalerary  day  in  every 
fourth  year.  This  day  was  added  to  the  month  of 
February  making  it  to  consist  of  29  instead  of  28  days. 
This  fourth  year  was  denominated  Bissextile^  and  is  now 
usually  called  Leajp  Year.     As  the  correction  should  have 


ART.  68.J  MEASURE    OF   TIME.  63 

been  5  hrs.  48  min.  and  51f  sec,  instead  of  6  hours,  by 
considering  every  fourth  year  as  consisting  of  866  days, 
there  was  involved  an  error  amounting  to  about  18  hours 
in  every  100  years  ;  to  remedy  which  every  one-hundredth 
year  was  considered  as  having  only  365  days.  But  the 
allowance  of  a  whole  day  in  every  100  years  was  too  much, 
by  nearly  one-fourth  of  a  day,  which  excess  in  every  400 
years  amounted  to  an  entire  day. 

Hence,  every  year,  (except  the  centennial  years,)  that 
is  divisible  by  4  is  a  Leap  Year,  and  every  centennial  year 
that  is  divisible  by  400  is  also,  a  Leap  Year.  The  next 
centennial  year  that  will  be  a  Leap  Year  is  2000. 

The  following  are  the  names  of  the  twelve  calendar 
months,  which  compose  the  civil  year,  and  the  number  of 
days  In  each : — 

Names.  Days. 


«  1 

1    1st  month  January, 

31 

1  1 

I    2d 

a 

February, 

28— in  leap  year  29. 

bD    1 

r  3d 

March, 

31 

•g     H 

4th 

April, 

30 

tc*  1 

[   5th 

May, 

-^ 31 

*-< 

r    6th 

June, 

30 

s 

7th 

July, 

31 

m   1 

[    8th 

August, 

31 

a   . 

{   9th 

September, 

30 

_s  ^ 

10th 

October, 

31 

<j   1 

[nth 

November, 

30 

^   1 

12th 

a 

December, 

31 

"  Thirty  days  hath  September, 
April,  June,  and  November; 
February  28,  alone, 
All  the  rest  have  thirty-one. 
Except  in  Leap  Year  ;  then  is  the  time 
When  February  has  twenty-nine." 

The  following  Table  will  enable  us  readily  to  determine 
the  number  of  days  from  one  date,  to  any  other  particular 
date  in  the  same  year : — 


64 


DENOMINATE    NUMBERS. 


[chap.    111. 


EXHIBITINO    THE    NUMBER    OF    DAYS    FROM    ANY    DAY    OF    OWE    MONTH    TO    THE    8AMI 
DAY    OF    ANY    OTHER    MONTH    IN    THE    SAME    YEAR. 


FROM  ANY 
DAY  OF 

to  the  same  day. 

Jan. 

Feb.;  Mar. 

April  May.  June. 

July. 

Aug. 

Sept.!  Oct. 

Nov. 

Dec. 

January,.... 

355 

31 

69 

90 

120  151 

181 

212 

243  ,  273 

304 

334 

February,.. 

334 

365 

28 

69 

89 

120 

150 

181 

212 

242 

273 

303 

March, 

306 

337 

365 

31 

61 

92 

122 

153 

184 

214 

245 

275 

April, 

275 

306 

334 

365 

30 

61 

91 

122 

153 

183 

214 

244 

May, 

•245 

276 

304 

335 

365 

31 

61 

92 

123 

153 

184 

214 

June 

214 

245 

273 

304 

344 

365 

30 

61 

92 

122 

153 

183 

July 

184 

215 

243 

274 

304 

335 

365 

31 

62 

92 

123 

163 

August 

153 

184 

212 

243 

273 

304 

334 

365 

31 

62 

92 

122 

September,.. 

122 

153 1 181 

212  1  242 

273 

303 

334 

365 

30 

61 

91 

October 

92 

123  1151 

182  212 

243 

273 

304 

335 

365 

31 

61 

November... 

61 

92  120 

161  181 

212 

242 

273 

304 

334  ]  365 

30 

December,... 

31 

62   90 

121  151 

182 

212 

243 

274 

304  1  335 

366 

How  many  days  from  the  12tli  of  May  to  the  12th  of 
October  ?  In  the  column  of  months  oq  the  left  of  the 
page  we  find  April; — passing  the  eye  along  the  horizontal 
row  of  figures  till  it  comes  to  the  perpendicular  column, 
headed  "  Oct.,"  we  find  153  days  to  be  the  time.  If,  in- 
stead of  "  12th  of  Oct.,"  in-  the  above  question,  we  sub- 
stitute the  20th  of  Oct.,  then  to  the  153  days,  add  the 
excess  of  20  above,  12  =  8  ;  and  we  have  153  +  8  =  161 
days,  for  the  number  of  days. 

When  there  are  29  days  in  February  the  proper  allow- 
ance must  be  made,  as  the  table  considers  28  days  in  Feb- 
ruary. 

Books. 


A  sheet  folded  in 


two  leaves  is  called  a  Folio. 
four      "       "      "a  Quarto  or  4to. 
eight      "      "      "  an  Octavo,  or  8vo. 
twelve      "      "      "  a  Duodecimo  or  12mo 
eighteeen      "      "      "  an  ISmo. 
twenty-four      "      "      "a  24mo. 


MISCELLANEOUS    TABLE. 


12  Units 
12  Dozen 
12  Grosd 
20  Units 


make 


1  Dozen. 
1  Gross. 
1  Great  Gross. 
1  Score. 


ART.    69.]       ADDITION    OF   DENOMINATE    NUMBERS. 


65 


24  Sheets  of  paper    make     1  Quire. 
20  Quires  "        1  Ream. 


30  Pounds 

make 

1  Bushel  of  oats. 

46  Pounds 

u 

1  Bush,  of  buckwheat  or  barley. 

56  Pounds 

(( 

1  Bushel  of  Indian  corn  or  rye. 

60  Pounds 

u 

1  Bushel  of  wheat. 

70  Pounds 

(( 

1  Bushel  of  salt. 

55  Pounds 

make 

1  Firkin  of  butter. 

196  Pounds 

(( 

1  Barrel  of  flour. 

200  Pounds 

(( 

1  Barrel  of  pork. 

200  Pounds 

u 

1  Barrel  of  beef. 

200  Pounds 

u 

1  Barrel  of  shad  or  salmon. 

14    Pounds  of  lead, 

or  iron 

make    1  Stone. 

21i  Stone 

"        1  Pig. 

8"  Pigs 

1  Fother. 

Addition  of  DENOMmATE  Numbers. 

Art.  69.  Addition  of  dcTiominate  numbers  is  finding 
the  sum  of  two  or  more  numbers  of  different  denominations 
in  the  same  kind  of  measure. 


1.  What  is  the  sum  of  £S 
15s.  lOd.,  and  Mi  10s.  3d.? 


16s.  5d.,  £S   12s.  9d.,  £Q 


Explanation. — Write  the   given    num- 
ers  of  the  same  denomination  one  under  an- 
other, and  place  12  over  the  d.  and  20  over 
the  8.  (to  cause  the  pupil  to  bear  in  mind 
that  12  pence  make  1   shilling,   and  20 
shillings  make  1  pound.)    Then,  commenc- 
ing at  the  column  of  pence,  we  add  it  as  in 
simple  addition,  and  obtain  for  the  sum  27 
pence.     In  27  pence,  how  many  shillings  1 
There  are  12  pence  in  Is.,  therefore,  one- 
twelfth    of  the  number  of  pence   equals 
the  number  of  shillings ; — 12  is  contained  in  27,  2  times  and 
3d.  remaining.     Place    the  3d.    under  the  column  of  pence, 
and  add  the  2s.  to  the  column  of  shillings,  and  we  obtain  for 


operation. 

20 

12 

£         8. 

d. 

8      16 

J 

3      12 

9 

6      15 

10 

4      10 

3 

23      15 


66  DENOMINATE    NUMBERS.  [cHAP.    III. 

the  sum,  55  shillings.  In  55s.,  how  many  pounds  ?  There 
are  2()s.  in  XI,  therefore,  one-twentieth  of  the  number  of 
shillings,  equals  the  number  of  pounds; — 20  is  contained 
in  55,  2  times  and  15s,  remaining.  Place  the  15s.  under 
the  column  of  shillings,  and  add  the  £2,  to  the  column  of 
pounds,  and  we  obtain  for  the  sum  £23.  This  we  place  under 
the  column  of  £,  and  we^iave  for  the  answer,  £23  15s.  3d, 


2. 

3. 

£ 

s. 

d. 

far. 

£ 

s. 

d. 

far. 

14 

10 

6 

2 

16 

11 

4 

1 

16 

15 

10 

3 

23 

13 

11 

2 

13 

12 

4 

1 

25 

15 

8 

0 

23 

9 

3 

0 

17 

18 

10 

3 

Troy   Weight. 


4, 

5. 

lb. 

oz. 

pwt. 

P^r. 

lb. 

oz. 

Pwt. 

Ri"- 

24 

7 

15 

20 

14 

7 

14 

20 

30 

9 

19 

13 

24 

9 

16 

16 

22 

8 

17 

14 

26 

10 

3 

18 

16 

11 

0 

12 

36 

11 

17 

15 

Avoirdupois    Weight.    - 

6.  7. 

T.     cwt.     qr.     lb.     oz.     dr.  T.     cwt.  qr.     lb.  oz.  dr. 

17  14   3  20  8  9  25  14  3  22  11  2 

2   8   1  15  10  7  35  19  1  21  10  8 

4  13   0  13  6  10  16  20  0  16  9  7 

6  16   2  12  11  8  17  21  1  19  1  9 


Apothecaries'  Weight. 


lb. 

:?. 

3. 

^. 

Rr. 

5 

1 

2 

1 

15 

8 

10 

7 

2 

19 

L4 

10 

2 

3 

7 

5 

1 

2 

3 

16 

7 

9 

3 

1 

12 

lb.  5-  3.    3.  gr. 

10  1  6  0  10 
0  1  2^  1  16 

13  2  7  3  20 

11  0  5  4  13 
17  2  4  2  18 


art.  69.]     additon  of  denominate  numbers.  §1 

Cloth  Measure. 


10. 

11. 

12. 

vd.    qr.    na. 

E.Fi 

•,  qr.  na. 

E.E     qr.  na. 

'5    3    3 

3 

4    3 

17    4    2 

8    0     1 

7 

5     0 

18     3     1 

7    2    2 

10 

2    1 

24     1     3 

8    3    0 

19 

3    2- 

17    2    0 

13    1 

23 

1     1 

15     0    2 

Long  Measure. 

13. 

14. 

m. 

fur. 

rd.    yd.    ft.    in 

deff.      m.  fur.    rd.    yd.    ft. 

in. 

91 

7 

29      4     2      10 

122    19 

4      15    0 

9 

13 

3 

27    0    1      9 

12      7 

3    23    0     1 

9 

7 

7 

39     1    2      4 

27'     4 

7    36     1    2 

11 

15 

6 

32    0    0      7 

32      1 

3    26    0    0 

10 

23 

7 

23    1    2      8 

34      0 

6     39     1     2 

8 

30 

5 

28    0    1      6 

L,    0 

17      4 

5     37    0     1 

7 

Superficia 

r  Square  Measure. 

15. 

S.  M.        A. 

R. 

p.      sq.yd.    sq.  ft.  sq.  in. 

123      465 

1 

27      20        6 

14 

12      121 

1 

12        8        3 

122 

36      376 

2 

32        7        7 

100 

37      468 

1 

17        2        6 

132 

19      478 

3 

12        0        8 

140 

17      300 

0 

34        0        5 

96 

Surveyor's  Measure. 

16. 

17. 

.  ^■ 

fur.  cha.  P.      li. 

m.    fur 

.  cha.  P.    li. 

8C 

)    2    4    2    11 

187    2 

2    19 

18 

!     6     3     1     21 

11    4 

9     3     17 

IS 

;     2    9     3     17 

24    3 

8     1     12 

2^ 

5    7    8    2    18 

36    1 

7     0    14 

13 

1    5     7    1     16 

41    0 

0     1     18 

11 

.    3    5     0    14 

73     7 

6     1    23 

68  DENOMINATE    NUMBERS.  [CHAP.    III. 

Solid,  or  Cubic  Measure. 


18. 

19. 

T.  cu.  ft. 

cu.  in. 

c. 

cu.  ft.  cu.  in. 

84  12 

1364 

12 

120  1463 

18  32 

1431 

12 

121  1612 

16  12 

931 

91 

112  1316 

30  28 

1246 

36 

97   362 

73  17 

863 

97 

88  1473 

96  14 

1241 

Wm 

E  Measure 

12 

12   784 

^ 

Jt 

20. 

21. 

Tun 

.  hhd.  gal.  qt. 

pt.  g^i. 

hhd.  gal.  qt. 

100  19  1 

pt 

94 

1  26  2 

1 

4 

1  21  3 

1  3 

12  37  3 

1 

12 

0  32  1 

0  2 

32  46  0 

0 

17 

1  47  2 

1  1 

14  37  2 

1 

26 

0  53  1 

0  0 

18  17  1 

1 

34 

0  60  1 

1  2 

OR 

Beer  Measur] 

22   6  2 

0 

Ale, 

E. 

22. 

23. 

hhd.  gal.  ( 

qt.  pt. 

bar 

.  gal.  q^t.  pt. 

39  31 

1  0 

49 

3  23 

2  1 

7 

32  3  1 

8  40 

3  0 

6 

27  1  0 

7  37 

1  1 

7 

13  2  0 

6  38 

2  1 

12 

18  3  1 

12  52 

3  1 

Dry  Measure 

14 

17  1  1 

24. 

25. 

ch.  bu.  pk. 

qt.  pt. 

bu.  pk.  qt.  pt. 

75   1  2 

2  1 

81  0  0  1 

12  35  3 

7  1 

23  3  7  1 

16  25  1 

5  0 

« 

12  0  1  0 

10  32  2 

4  0 

17  2  6  0 

11  17  1 

6  1 

10  3  5  1 

22  34  0 

3  1 

16  1  4  1 

ART.  70.]       SUBTRACTICN    OF   DENOMINATE    NUMBERS. 


69 


26. 


0. 

s. 

• 

/ 

// 

21 

0 

15 

38 

5 

1 

7 

12 

40 

32 

2 

9 

17 

35 

16 

3 

8 

23 

37 

46 

3 

10 

29 

57 

54 

8 

11 

21 

46 

3Y 

28. 

Me. 

yr. 

da. 

hr. 

min. 

sec. 

99 

155 

1 

50 

44 

12 

10 

13 

42 

27 

16 

102 

18 

24 

36 

19 

8 

21 

54 

57 

23 

13 

19 

49 

48 

29 

18 

23 

58 

56 

Circular  Measure. 


0. 


Measure  of  Time. 


27. 


17  11  19  53  11 

1  3  21  39  52 

2  9  13  42  47 
7  -6  27  55  19 
1  11  28  18  17 
4  3  18  16  56 


29. 


wk.  da.  hr.  min.  sec. 


49 

3 

19 

36 

5 

3 

4 

20 

42 

17 

1 

6 

21 

47 

39 

2 

2 

19 

58 

52 

17 

4 

16 

48 

58 

23 

5 

12 

18 

19 

Subtraction  of  Denominate  Numbers. 

Art.  70.  Subtraction  of  denominate  numbers  is  finding 
the  difference  between  two  denominate  numbers. 

1.  From  ieSB  8s.  lOd.  1  far.,  take  £12  15s.  4d.  3  far. 

Explanation. — Write  the  subtrahend 
under  the  minuend,  observing  to  place 
the  numbers  of  the  same  denomination 
one  under  another ;  and  begin  at  the 
right  to  subtract.  We  cannot  take  3 
farthings  from  1  far.,  therefore,  from 
the  lOd.  (of  the  minuend)  we  take  1 
penny,  which  equals  4  far.,  and  add  it 
to  1  far.,  which  makes  it  5  farthings ;  3  far.  from  5  far.  leave 
2  far.,  which  place  under  the  column  of  farthings.  We  now  take 
the  4d.  from  the  9d.,  or  add  1  penny  to  the  4d.,  and  take  this  sum 
from  lOd.,  which  in  either  case  gives  the  same  remainder,  5d. 
As  we  cannot  take  15s.  from  83.,  we  borrow  £1,  =  20s.,  from 
the  £38,  and  add  it  to  the  8s. ;  from  this  sum,  we  take  the  15s., 
and  obtain  13s.  for  a  rema  nder.    We  now  take  £12  from  £37, 


OPERATION. 

20 

12 

>  4 

£. 

s. 

d. 

far. 

Min. 

38 

8 

10 

1 

Sub. 

12 

15 

4 

3 

Rem.    25     13      5    2 


TO  DENOMINATE    NUMBERS.  [cHAP.    III. 

or  add  £1  to  the  £12,  and  take  the  sum  from  the  £38,  which 
in  either  case,  gives  the  same  remainder,  £25,  Hence  the  dif- 
ference of  the  two  quantities  is  £25  13s.  5d.  2  far. 

2.  3. 

£        s.       d.  far.  lb.     oz.  pwt.    er. 

24      8    6    2  25    8    17    21        *> 

16    12    7    3  •  14    7    18    23 


4.  5. 

T.    cwt.    qr.    lb.  oz.  dr.  T.  cwt.     qr.  lb.  oz.    dr. 

50  16  1  23  IQ  12  14  10  2  12  4  8 

27   7  3  24  3  14  5  14  3  20  7  12 


6. 

7. 

lb. 

14 

3.  3.  3 
11  6  1 

>.  gr. 

12 

yd. 
18 

qr.  na 

2  1 

8. 

12 

10  3  2 

1 

5 

12 

3  3 

9. 

10. 

E.Fr, 

14 

■  t 

na. 

2 

E.E. 

19 

^2 

na 
1 

3eg. 

21 

m. 
2 

fur.  rd. 

7  21 

yd.  ft.  in 

3  18 

10 

5 

3 

16 

4 

3 

18 

45 

3  25 

4  2  10 

11. 

12. 

ni. 
16 

fur. 

1  5 

cha.  P. 

3  1 

lin. 
10 

36 

[.  A.   R.   P. 

276  2  12 

sq.^yd. 

13 

2 

8  2 

12 

24 

108  3  37 

251 

13. 

14. 

T. 

16 

en. ft.  cu.ir 

32  1421 

1. 

c. 
16 

cu  ft.   cu.in. 

110  1612 

6 

37  1675 

11 

116  1719 

15. 

16. 

. 

Tnn. 
32 

hhd. 
0 

gal.  qt. 

24  3 

^(5- 

t 

hhd.  gal. 

62  41 

8^-5* 

15 

2 

52  3 

1 

3 

49  60 

2  1 

4.RT.    70.]      QUESTIONS   IN   DENOMINATE    NUMBERS.  *ll 


17.   " 

ch.  bu.  pk.  qt.  pt. 

30  12  3  3   1 
17  30  3'  7   0 

wk. 

36 
26 

bn. 

27 
9 

18. 

3  7 

1 

19. 

C.        S.       "        '        " 

20  10  15  24  32 
9   5  24  56  52 

da. 

3 
6 

20. 

hr.  min 

12  43 

20  55 

sec. 

15 
32 

Practical  Questions  in  Addition  and  Subtraction  of 
Denominate  Numbers, 

1.  Erom  a  piece  of  cloth  containing  27  yards  3  qrs. 

1  na.,  there  were  taken  three  garments;  the  first  contain- 
ed 8  yds.  8  qrs.  2  nas. ;  the  second  4  yds.  1  qr.  8  nas. ;  and 
the  third  2  yds.  8  qrs.  8  nas.; — how  much  remained  ? 

2.  Bought  a  hogshead  of  sugar  wieghing  9  cwt.  3  qrs. 
21  lbs.;  sold  to  A  1  cwt.  2  qrs.  15  lbs.;  to  B  2  cwt.  8  qrs. 
24  lbs.;  and  to  C  3  cwt.  1  qr.  15  lbs.; — how  much  re- 
mained unsold  ? 

3.  A  man  agrees  to  build  186  rods  and  15  feet  of  stone 
fence; — at  one  time  he  built  86  rds.  2  feet;  at  another 
time  56  rds.  8  feet;  and  at  another  time  10  rds.  1  foot. 
How  much  still  remains  to  be  built, 

4.  I  agreed  to  let  a  person  have  24  T.  9  cwt.  2  qrs. 
15  lbs.  of  hay.  He  took  away  four  loads,  the  weight  of 
which  were  as  follows:  the  first  weighed  16  cwt.  2  qrs. 
18  lbs.;  the  second,  19  cwt.  3  qrs.  12  lbs.;  the  third,  1  T. 

2  cwt.  1  qr.  21  lbs.;  and  the  fourth,  1  T.  5  cwt.  2  qrs. 
14  lbs. ; — to  how  much  hay  is  he  still  entitled  ? 

5.  How  many  yard  of  cloth  in  three  pieces:  the  first 
containing  12  yds.  3  qrs.  2  nas.;  the  second  6  E.  English 
2  qrs.  1  na.;  the  third  9  E.  French,   1  qr.  8  nas.? 

6.  Bought  three  pieces  of  cloth:  the  first  containing  25 
yds.  8  qrs.  1  na.;  the  second  47  yds.  1  qr.  8  uasrj  and  the 
third  85  yds.  8  qrs.  2  nas.; — I  sold  73  yds.  8  qrs.  2  nas.  of 
it.     How  much  remained  unsold  ? 


T2  DENOMINATE    NUMBERS.  [CHAP.    III. 

T.  A  merchant  bought,  at  one  time  956  bnshels  and  3 
pecks  of  Indian  corn;  at  another  time  '759  bushels,  2  pks. 
and  t  quarts;  and  sold  325  bush.  3  pks.  and  6  qts.  of  it. 
How  much  had  he  remaining  ? 

8.  John  is  23  years,  9  month,  and  18  days  old;  James 
is  18  years,  10  months,  and  25  days  old.  What  is  the  dif- 
ference of  their  ages  ? 

9.  Suppose  a  person  was  born  February  29,  1*188;  how- 
many  birth-days  will  he  have  seen  'on  February  29,  1840, 
not  counting  the  day  on  which  he  was  born  ? 

10.  A  merchant  sold  goods  to  the  amount  of  iS39t  18s. 
6d.  2  qrs.;  and  received  in  payment  £199  19s.  lOd.  3  qrs.; 
how  much  remains  due  ? 

11.  From  a  pile  of  wood  containing  423  cords,  Isold  at 
one  time,  56  C.  112  cu.  ft.;  at  another  time,  91  C.  113  cu. 
ft.;  at  another  time,  126  C.  96  cu.  ft.  How  many  cords 
remain  unsold  ? 

12.  How  long  from  the  birth  of  William  Shakspeare, 
April  23,  1564,  to  the  birth  of  Milton,  Dec.  9,  1608  ? 

13.  A  farmer  raises  125  bush.  2  pks.  6  qts.  of  wheat  on 
one  field;  19t  bush.  1  pk.  T  qts.  on  another  field:  he  sells 
to  one  person  97  bush.  3  pks.  7  qts.;  and  to  another  per- 
son 112  bush.  2  pks.  6  qts.  How  many  bushels  has  he 
remaining  ? 

14.  A  gentleman  owned  three  tracts  of  land:  the  first  of 
which  contained  127  acres,  3R.  l5  rods;  the  second,  496  A. 
1  R.  25  rods;  the  third,  525  A.  0  R.  35  rods;  how  much 
remained  after  he  sold  1008  A.  2  R.  25  rods? 

15.  Suppose  a  note  given  Sept.  10,  1796,  to  be  paid 
March  5,  1808.  How  long  was  the  note  on  interest,  if  we 
count  30  days  to  the  month  ?  How  long  if  the  time  is 
accurately  computed  ? 

Multiplication  of  Denominate  Numbers, 

Art.  71.  Multipligation  of  denominate  numhers  is  tak- 
ing a  quantity  of  different  denominations  as  many  times  as 
there  are  units  in  another  number. 

Multiply  £5  12s.  6d.  by  5.  , 


ART.  11.]    MULTIPLICATION  OF  DENOMINATE  NtJMBERS.  73 

OPERATION.      Explanation The    numbers    being    properly 

&     s.    d.    written  down,  we  begin  at  the  right  to  multiply. 

5  12  6     5  times  6d.  are  30d.,  in  30d.  how  many  shillings  ? 

5     There  are  12d.  in  Is.,  therefore,  one-twelfth  of  the 

number  ofpence  equals  the  number  of  shillings.  12  is 

28     2  6     contained  in  30,  2  times,  and  6d.  remaining  \ — write 

the  6d.  under  pence,  and  reserve  the  28.     5  times 

I2s.  are  60s.,  and  2s.  added,  are  62s.,  which  equals  £3  2s. ; — 

write  the  2s.  under  shillings,  and  reserve  the  £3.     5  times 

£5  are  £25,  and  £3  added  are  £28.     Hence,  &c. 


2. 

» 

3. 

4. 

£        s.   d. 

far. 

cwt.  gr.  lb.   oz. 

22  3  21  12 

T. 

cwt.  qrs. 

lb. 

oz. 

12  10  8 

3 

4 

12  1 

20 

12 

4 

5 

7 

5. 

6. 

7. 

m,  fur.  rd. 

ft. 

deg.  m.  fur 

18  21  4 

rd. 

yds. 

qrs. 

nas 

12  7  32 

2 

20 

14 

3 

2 

12 

8 

14 

8. 

9. 

10. 

yds.  qrs.  na. 

cwt. 

qr.  lb. 

3  23 

T. 

cwt. 

qr.  lb. 
3   21 

oz. 

dr. 

17  3  1 

18 

4 

17 

12 

14 

20 

15 

9 

11.  How  much  cloth  will  it  take  for  9  suits  of  clothes, 
if  each  suit  require  8  yds.  2  qrs.  2  nas.  ? 

12.  How  long  will  it  take  a  man  to  chop  14  cords  of 
wood,  if  it  take  him  7  hours,  40  minutes,  and  50  seconds 
to  chop  1  cord  ? 

13.  What  is  the  weight  of  12  hogsheads  of  sugar,  each 
weighing  8  cwt.  3  qrs.  23  lb.  ? 

14.  If  a  span  of  horses,  at  1  load,  can  draw  1  cord  212 
cubic  feet  of  wood,  how  many  cords  can  they  draw  in  14 
loads  ? 

15.  If  a  family  of  6  persons,  consume  10  gallons,  3 
quarts,  and  1  pint  of  molasses  in  1  week;  what  quantity 
will  a  family  of  double  the  number  of  persons  consume  in 
1  year? 

4 


14  DENOMINATE   NUMBERS.  [CHAP.    Ill 

16.  What  is  the  weight  of  18  silver  spoons,  if  each 
weigh  5  oz.  14  pwt.  20  grs.  ? 

17.  If  1  acre  of  land  produce  45  bush.  3  pks.  T  qts. 
1  pt.  of  wheat,  how  much  will  12  acres  produce  ? 

18.  If  a  man  walk  25  miles,  5  fur.  2t  rds.  in  1  day,  how 
far  can  he  walk  in  9  weeks,  not  counting  Sunday  ? 

19.  An  estate  of  i£3295  15s.  6d.  is  divided  among  four 
children:  the  first  has  £125  16s.  lid.;  the  second  twice 
as  much,  lacking  ^802  18s.  9d.;  the  third  £M6  its.  9d.; 
and  the  fourth  the  remainder.  How  much  did  the  fourth 
receive  ?  • 

20.  If  a  locomotive  move  1  m.  25  rds.,  in  1  minute;  how 
far  will  it  move  in  1  day  ? 

Division  of  Denominate  Numbers. 

'  Art.  TS.  Division  of  denominate  numhers  is  the  process 

of  finding  any  proposed  part  of  a  given  number,  composed 

of  two  or  more  denominations  of  the  same  kind  of  measure. 

1.  If  5  barrels  of  sugar  weigh  9  cwt.  1  qr.  10  lbs.,  how 

much  will  1  barrel  weigh  ? 

OPERATION.  Explanation. — Write  the  divisor  on  the  left  of 
cwt.  qr.  lb.  the  dividend,  as  in  division  of  abstract  numbers 
5)9     1     10     5  is  contained  in  9,  once  and  4  cwt.  remaining. 

4  cwt.  =  16  qrs.,  to  which  add  the  1  qr.  and  it 

1     3     12     equals  17  qrs.,  5  is  contained  in  17,  3  times,  and 

2  qrs.  remaining.     2  qrs.  =  50  lbs.,  to  which 

add  the  10  lbs.,  and  it  =  60  lbs.     5  is  contained  in  60,  12  times. 

Therefore,  one-fifth  of  9  cwt.  1  qr.  10  lbs.,  is  1  cwt.  3  qrs.  12  lbs. 

Note. — It  is  impossible  to  divide  one  concrete  number  by  another,  (See 
Art.  44)  hence  in  the  above  example  we  do  not  divide  9  cwt.  1  qr.  10  lbs.  by 
6  barrels,  but  we  separate  the  9  cwt.  1  qr.  10  lbs.  into  5  equal  parts  j  the  6 
barrels,  being  considered  an  abstract  number. 


2. 

3 

4 

.. 

£ 

8. 

d.    iar. 

cwt. 

qr. 

lb. 

oz. 

T. 

cwt. 

I 

lb. 

)62 

7 

9    3 

5. 

6)101 

1 

13 

8 

7)32 
5. 

14 

15 

m.      fur. 

rd.    ft. 

deff. 

m.    fur 

.   rd. 

12)118    1 

0    12 

9)144 

26    4 

25 

8. 

yds. 

ft. 

in. 

7)196 

2 

11 

ART.    72.]         DIVISION  OF  DENOMINATE  NUMBERS.  75 


T.    cwt.  qr.    lb.    oz.    dr. 

9)44    1    1    1    3    14 


9.  10. 

A        R       p  T.     cwt.   qr.     lb.      oz.      dr. 

11)346    3    37  5)19     18    3    20    12    13 


11.  Divide,  ^£346  18s.  4d.  2  far.  by  47. 

OPERATION. 

20        12         i,y 

£  s.       d.     far. 

47)346    18    4    2    (£7  7s.  7d.  2  far.    Ans. 
329 

17 

20 

47)  358  (7s. 
329 


29 
12 


47)352(7d. 
329 


23 
4 


47)94(2  far. 
94 


.0 

12.  Divide  137  lbs.  9  oz.  18  pint.  19  grs.  by  23. 

13.  If  451  individuals  share  equally  8021  T.  12  cwt. 
1  qr.  6  lbs.  8  oz.  of  sugar,  how  much  will  each  receive  ? 

14.  If  13  hogsheads  of  sugar  weigh  6  T.  8  cwt.  2  qrs.  7 
lbs.;  how  much  will  1  hogshead  weigh  ? 

15.  A  vintner  sold  33  hhds.  56  gals,  of  wine,  to  15  dif 
ferent  men ;  how  much  did  each  buy,  providing  they  each 
purchased  an  equal  quantity  ? 

16.  If  a  man  travel  348  mi.  1  fur.  12  rds.  in  28  days, 
how  far,  on  an  average  is  that  a  day  ? 


'^6  DENOMINATE    NUMBERS.  [cHAP.    Ill 

IT.  A  merchant  sold  320  yds.  2  qrs.  2  nas.  of  broad- 
cloth, in  19  successive  days;  how  much  did  he  sell  daily, 
providing  he  sold  the  same  quantity  each  day  ? 

18.  A  produce  dealer  divided  132  bushels,  3  pks.  t  qts. 
of  wheat,  equally  among  23  of  his  poor  neighbors;  how 
much  did  each  receive  ? 

19.  26  men  bought  645  acres  20  P.  of  land,  and  are 
to  share  it  equally;  how  much  ought  each  to  receive  ? 

20.  A  speculator  bought  H9  cwt.  1  qrs.  3  lb.  of  sugar, 
and  sold  it  to  36  men;  how  much  did  each  receive,  pro- 
viding each  bought  the  same  quantity  ? 


Practical  Questions  combining  Addition,  Subtraction, 
Multiplication  and  Division  of  Denominate  Num- 
bers. 

1.  A  farmer  having  19  cwt.  2  qrs.  19  lbs.  of  pork, 
sold  5  cwt.  3  qrs.  1  lb.  of  it,  and  the  remainder  he  put 
into  6  barrels;  how  much  did  each  barrel  contain  ? 

2.  Bought  of  A  91  acres,  2  R.  12  P.  of  land,  of  B 
4  times  as  much,  lacking  *I  acres,  1  R.,  and  of  C  one-half 
as  much  as  of  A  and  B  together ;  how  much  did  I  buy  of 
B  and  C  respectively,  and  how  much  in  all  ? 

3.  A  merchant  bought  9  pieces  of  silk,  each  contain- 
ing 5t  yds.  3  qrs.  Having  sold  to  another  merchant  one- 
third  of  it,  and  to  4  ladies,  each  9  yds.  3  qrs.  3  nas,,  how 
much  remains  unsold  ? 

4.  A  farmer  has  three  fields  of  wheat:  from  the  first 
he  obtains  224  bus.  2  pks.  2  qts.;  from  the  second  one- 
half  as  much,  increased  by  *I6  bus.  3  pks.  1  qt.;  and  from 
the  third,  as  much  as  from  the  other  two,  lacking  84  bus. 
2  pks.  1  qts.  How  much  did  he  obtain  from  the  three 
fields? 

5.  From  one-half  of  a  piece  of  cloth  containing  82  yds. 
2  qrs.,  a  tailor  cut  six  suits  of  clothes.  How  much  did 
each  suit  contain  ? 

6.  A,  B,  C,  and  D,  having  4  cwt.  3  grs.  20  lbs.  of 
sugar,  agree  to  divide  it  as  follows:  A  is  to  take  15  lbs 


ART.    72.]  PRACTICAL    QUESTIONS.  11 

and  one-fifth  of  the  remainder;  B  1  qr.  3  lbs.  and  one- 
fourth  of  the  remainder;  C  1  qr.  12  lbs.  and  one-third  of 
the  remainder;  and  D  is  to  have  what  now  remains.  How 
much  sugar  should  each  receive  ? 

I.  A,  B,  C,  and  D,  share  840  bushels,  3  pks.  of  wheat 
as  follows:  A  takes  16  bush.  3  pks.  and  one-fourth  of  the 
remainder;  B  takes  14  bush.  2  pks.  and  one-third  of  what 
remains;  C  takes  13  bush.  2^  pks.  and  one-half  of  what 
remains;  and  D  takes  what  now  remains.  How  much 
does  each  receive  ? 

8.  Bought  of  one  man  8  bus.  2  pks.  3  qrs.  of  grass- 
seed;  of  another  man  3  times  as  much,  and  2  bus.  2  qts. 
more;  and  of  another  3  times  as  much  as  of  the  second, 
lacking  1  bus.  1  pk.  6  qts.  How  much  did  I  buy  of  each 
respectively,  and  how  much  of  all  ? 

9.  32  men  agree  to  construct  28  miles.  4  fur.  32  rds.  of 
road; — after  completing  one-half  of  it,  one-fourth  of  the 
number  of  men  left  the  company.  What  distance  did 
each  man  construct  before  and  after  one-fourth  of  the  men 
left  ? 

10.  A,  B,  C,  and  D,  having  184  bus.  2  pks.  of  wheat, 
agree  to  divide  it  as  follows:  A  is  to  have  one-half  of  the 
whole;  B  is  to  have  one-third  of  the  remainder;  C  is  to 
have  one-fourth  of  what  then  remains;  and  D  is  to  have 
what  is  left.     What  is  the  portion  of  each  ? 

II.  Divide  448  acres,  3  R.  24  P.  of  land  among  A,  B, 
C,  and  D,  so  that  A  shall  have  one-eighth  of  the  whole, 
-\-  4  acres,  3  rds.  G  pis. ;  B  one-fifth  of  the  remainder ;  C 
one- third  of  what  then  remains;  and  D  the  rest.  How 
much  will  each  one  have  ? 

12.  An  estate  of  iE2490  is  to  be  divided  among  a  widow, 
two  sons,  and  three  daughters; — the  widow  receives  one- 
third  of  the  whole,  lacking  ^£34  6;  the  youngest  sou  re- 
ceives as  much  as  the  widow,  -f  £212;  the  oldest  son  re- 
ceives as  much  as  the  widow  and  youngest  son  together, 
lacking  £335  10s.;  and  the  three  daughters  share  equally 
of  the  remainder.     How  much  does  each  receive  ? 


IS  RKDCCTION.  [chap.   III. 


REDUCTION. 

-Art.  7*3.  If  the  quantity  is  to  be  changed  from  a 
higher  to  a  lower  denomination,  the  process  is  called  Re- 
duction Descending; — if  from  a  lower  to  a  higher  denom- 
ination, Reduction  Ascending. 

Reduction  Descending. 
1.  In  JE34  155.  Qd.  how  many  pence  ? 

OPERATION.  Explanation. — There  are  20a.    in  £1; 

20       12  therefore,  20  times  the  number  of  pounds 

£       8.       d.  equal  the  number  of  shillings.  20  times  34 

34      15       6  are  680,  and  15s.  added  =  695s.  There  are 

20  12d.  in  Is.;  therefore,  12  times  the  number 

of  shillings  equal  the  number  of  pence. 

o^^  12  times  695  are  8340,  and  6d.  added  =: 

12  8346d.    TherefoK  ^34  158.  6d.  =8346d. 


8346 


2.  In  £23  12s.  8d.  3  far.,  how  many  farthings  ? 

3.  In  18  lbs.  6  oz.  15  pwt.  14  grs.,  how  many  grains  ? 

4.  How  many  grains  in  1  lb.  1  5  2  9  12  grains  ? 

5.  How  many  drams  in  1  T.  3  qrs.  21  lbs.  10  drs.  ? 

6.  Reduce  14  cwt.  2  qrs.  20  lbs.  to  pounds. 
1.  Reduce  16  yards,  2  qrs.  2  nas.  to  nails. 

8.  In  12  E.  Fr.  5  qrs.  1  na.,  how  many  nails  ? 

9.  In  8  E.^.  4  qrs.  3  nas.,  how  many  nails  ? 

10.  In  1  mile,  how  many  feet  ? 

11.  In  1  mi.  5  fur.  35  rds.  5  yds.  2  ft.  6  in.;  how  many 
inches  ? 

12.  How  many  square  poles  in  102  acres,  3  R.  2t  P.? 

13.  In  5  hhd.  20 gals.  3  qts.  1  pt.;  how  many  pints? 

14.  In  2  pi.  2  gals,  2  gills;  how  many  gills  ? 

15.  In  6  barrels,   25  gals.  3  qts.   1  pt.  of  beer;    how 
many  pints  ? 

16.  In  8  bushels,  2  pks.  5  qts.  1  pt. ;  how  many  pints  ? 

17.  In  2  weeks,  5  days,  5  hours,  and  5  minutes;  how 
many  minutes  ? 


ART.    73.]  REDUCTION    ASCENDING.  79 

18.  In  1  day,  how  many  minutes  and  seconds? 

19.  In  1  year,  how  many  hours. 

20    Tti  5  days,  4  hours,  45  seconds;  how  many  seconds ? 

21.  In  1  T.  1  lb.  1  dr.;  how  many  drams  ? 

22.  In  1  acre;  how  many  square  feet  ? 

Reduction  Ascending. 

1.  In  647d.,  how  many  pounds,  shillings  and  pence  ?  ' 

OPERATION.  Explanation. — ^There  are  12  pence  in 

d.  Is.;  therefore,  one-twelfth  of  the  num- 

12)647  ber  of  pence  equals  the  number  of  shil- 

lings,  which  is  53s.,  and  lid.  remaining. 

20)53    lid.  rem.  In  53s.  how  many  pounds  7     There  are 

20s.  in  £1 ;  therefore  one-twentieth  of 

2    13s.  lid.      the  number  of  shillings  equals  the  num- 
ber of  pounds,  which  is  £2,  and  13s. 

remaining.     Therefore,  647d.  =  £2  13s.  lid. 

2.  In  16823  far.,  bow  many  pounds,  shillings,  &c.  ? 

3.  In  84672  grs.  Troy  Weight;    how  many  pounds, 
ounces  &c.  ? 

4.  How  many  pounds,  &c.,  Apothecaries'  Weight,  in 
569  5? 

5.  In   1894763  dr.  Avoidupois  Weight;    how  many 
tons,  cwt.  &c.  ? 

6.  In  89643  lbs.;  how  many  tons,  cwt.  &c.  ? 

7.  In  8467  nails;  bow  many  yards,  qrs.  &c.  ? 

8.  In  2706  nas.;  how  many  E,  E.,  qrs.  &c.  ? 

9.  In  4762  nas,;  how  many  E.  Fr.,  qrs.  &c.  ? 

10.  In  84672  feet;  how  many  miles,  &c.  ? 

11.  In  3647  rods;  how  many  miles,  furlongs,  &c.  ? 

12.  In  1478  P.;  how  many  acres,  roods  and  poles  ? 

13.  In  165  qts.;  how  many  gallons  ? 

14.  In  20042  gills;  how  many  hogsheads,  &c.  ? 

15.  In  17632  gallons;  how  many  tons,  &c.  ? 

16.  In  4007  pints  of  beer;  how  many  barrels,  &c.  ? 

17.  In  147  pints;  bow  many  bushels,  &c.  ? 

18.  In  64  pints;  how  many  bushels  .? 

19.  In  86400  seconds;  how  many  days  ? 

20.  In  3146232  secondsj  how  many  weeks,  days,  &c  ?' 


80  PROPERTY   OF  THE   NUMBER   9.  [cHAP.    IV 

CHAPTER  TV. 

Peculiar  Property  of  the  I^umber  9. 

Art.  74.  Any  number  is  divisible  by  9,  when  t/ie  sum 
of  its  digits  is  divisible  by  9.  Consequently,  every  number 
divided  by  9,  will  give  the  same  remainder  as  the  sum  of  its 
digits  divided  by  9. 

Also,  if  from  any  number,  the  sum  of  its  digits  be  sub- 
tracted, the  remainder  will  be  divisible  by  9. 

Note.     The  pro6r  of  the  fundamental  Rules  of  Arithmetic,  is  founded  upon 
the  above  properties  of  the  number  9,  which  we  will  now  consider. 

Take  any  fiumber,  as  t65,  which  equals  TOO  +  60  +  5. 

Now,  100  =  tXl00  =  Tx  (99  +  1)      =      1  X  99  +  t 

60  =  6X.  10=6X(  9  +  1)      =      6X    9  +  6 

5=  5 

Hence,   165=  1X99  +  6x9  +  1  +  6  +  5 

But,  1  X  99  +  6  X  9,  which  lacks  the  sum  of  the 
digits  of  the  number,  165,  of  being  equal  to  that  number, 
is  divisible  by  9;  since  each  of  the  expressions,  1  X  99 
and  6x9,  contains  the  factor  9.  Hence,  if  the  remain- 
ing part  of  the  number,  which  is  the  sum  of  its  digits,  is 
divisible  by  9  the  number  itself  is  divisible  by  9.  As 
every  number  can  be  separated  into  two  parts, — the  sum 
of  its  digits,  and  another  number,  divisible  by  9,  it  follows 
that  the  same  remainder  will  be  found  by  dividing  the 
number  by  9,  as  is,  by  dividing  its  digits  by  9:  Also,  if  a 
number  be  diminished  by  the  sum  of  its  digits,  the  remain- 
der will  be  divisible  by  9. 

Multiplication  of  Abstract  Polynomials. 

1.  Multiply  5  +  1  by  3  +  6. 

operation.  Explanation Coinmence  at 

517  the  left,  and  multiply  each  terra 

3  I  g  in  the  multiplicand  successively, 

by  each  term  in  the  multiplier. 


15  -L  21  +  30  4  42  Product.     It  is  evident  that  the  sum  of  these 


ART.  82.]  DEFINITIONS.  81 

several  partial  products,  (15  -}-  21  -f-  30  -f-  42 :=  108,)  is  equal 
to  the  product  of  the  sum  of  5  and  7  by  the  sum  of  3  and  C. 

2.  Multiply  2  +  3  +  4  by  4  +  6  +  1. 

3.  Multiply  8  +  6+2  by  2  +  3  +  4. 

4.  Multiply  4  +  6+^  +  8  by  3  +  2  +  4. 

5.  Multiply  1  +  2  +  3  +  4  +  5  +  6  by  t  +  8  + 9. 

6.  Multiply  9  +  8  +  ti- 6  +  5  +  4  by  1  +  2  +  3-1-  4. 


Miscellaneous  Definitions. 

Art.    75.  An  Integer  is  any  whole  number. 

Art.  76.  An  Even  number  is  any  integer  that  con- 
tains 2  a  whole  number  of  times,  without  a  remainder. 

Art.  77.  An  Odd  number  is  any  integer  that  does  not 
contain  2  a  whole  number  of  times  without  a  remainder. 
Hence,  an  odd  number  differs  from  an  even  number  by  a 
unit. 

Art.  78.  A  Prime  number  is  any  integer  that  cannot 
be  produced  by  multiplying  two  numbers  together,  each 
of  which  is  greater  than  a  unit;  as  1,  2,  3,  5,  7,  11, 13,  17, 
19,  23,  &c. 

Art.  79.  A  Composite  number  is  an  integer  that  can 
be  produced  by  multiplying  two  numbers  together,  each 
of  which  is  greater  than  a  unit.  Thus,  48  is  a  composite, 
number  and  may  be  produced  either  by  multiplying  to- 
gether the  composite  factors,  6  and  8,  or  the  prime  factors 
2,  2,  2,  2  and  3. 

Art.  80.  The  Prime  factors  of  a  number  are  the  prime 
numbers  that  are  multiplied  together  to  produce  that  num- 
ber.    The  prime  factors  of  48  are  2,  2,  2,  2,  and  3. 

Art.  8 1 .  All  integers  are  prime  numbers  or  composite 
numbers ;  and  all  composite  numbers  are  composed  of 
prime  factors  ; — hence  every  integer  is  a  prim£  number  or 
composed  of  prime  factors. 

Art.  82.  A  Square  number  is  a  composite  number, 
which  is  composed  of  two  equal  factors ;  as  9,  (=3x3)  ;  25, 
(«6X5);   49,    (=7X7)    &c. 

4* 


82  PRIME    NUMBERS.  [cHAP.    IV 

Art.  83.  A  Cube  number  is  a  composite  number,  that 
is   composed    of  three  equal  factors;  as    8,  (-=2x2x2);  2t 

(=3X^X3)    &c. 

Art.  84.  The  symbol    .*. ,  is  equivalent  to  the  word 
therefore  or  consequently. 


PRIME  NUMBERS. 

Art.  85.  All  prime  numbers,  except  the  digit  2,  are  odd 
numbers;  consequently,  they  terminate  with  an  odd  digit; 
as,  1,  3,  5,  *I,  or  9.  All  numbers  that  end  in  5  are  divisible 
by  5,  since  the  remainder,  (if  any  next  preceding  the  5,) 
is  a  certain  number  of  tinges  10,  which  added  to  5,  gives  a 
number  divisible  by  5,  since  each  of  the  numbers  composing 
it,  contains  the  factor  5; — therefore,  all  prims  numbers^ 
except  2  and  5  must  terminate  with  1,  3,  T,  or  9. 

Hence,  to  determine  whether  a  given  number  is  a  prime, 
first,  inspect  its  terminating  figure,  and  if  it  differs  from 
1,  3,  t,  or  9,  it  is  a  composite  number;  if  not,  it  may  still 
be  conposite  ; — now,  if  we  can  find  no  number  between  2, 
and  another  prime,  the  square  of  which  is  not  less  than  the 
given  number,  that  will  divide  it,  the  number  is  a  prime. 

Art.  86.  An  odd  number  divided  by  an  even  number 
gives  an  odd  number  for  a  remainder;  hence,  if  any  pme 
number,  except  2  and  3,  be  divided  by  6  the  remainder 
will  be  1,  3,  or  5  ;  but,  the  remainder  cannot  be  3,  as  the 
number  would  then  have  been  divisible  by  3,  since  the 
divisor  and  remaiiider  are  each  divisible  by  3.  Therefore, 
any  prime  number,  except  2  and  3,  when  divided  by  6  will 
give  1  or  b  for  a  remainder. 

The  following  table  is  sufficiently  extended  for  ordinary 
calculations. 


ART.   86.] 


PRIME    NUMBERS. 


ss 


Table  of  Prime  Numbers. 


1 

163 

383 

|6iU 

'  877lll^9 

1433 

1697 

1999 

2293 

2609 

2 

167 

389 

631 

1  88111151 

1439 

1699 

2003 

2297 

2617 

3 

173 

397 

641 

883 

1153 

1447 

1709 

2011 

2309 

26211 

5 

179 

491 

643 

887 

1163 

1451 

1721 

2017 

2311 

2633 

1 

181 

409 

647 

907 

1171 

1453 

1723 

2027 

2333 

2647 

.    11 

191 

419 

653 

911 

1181 

1459 

1733 

2029 

2339 

2657  i 

!  13 

193 

421 
43f 

659 

919 

118711471 

1741 

2039 

2341 

2659  1 

IT 

197 

661 

929 

1193 

UM 

1747 

2053 

2347 

2663 

19 

199 

433 

673 

937 

1201 

14.53 

1753 

2063 

2351 

2671 

28"211 

439 

677 

941 

}^IS 

1487 

1759 

2069 

2357 

2677 

29 

223 

443 

683 

947 

1217 

1489 

1777 

2081 

2371 

26S3 

•  31 

227 

449 

691 

953 

1223 

1493 

1783 

2083 

2377 

2687' 

37 

229 

457 

701 

967 

1229 

1499 

1787 

2087 

2381 

2689  j 

41 

233 

461 

709 

971 

1231 

1511 

1789 

2089 

2383 

2693  i 

43 

239 

463 

719 

977 

1237 

1523 

1801 

2099 

2389 

2699' 

47 

241 

467 

727 

983 

1249 

1531 

1811 

2111 

2393 

2707; 

53 

251 

479 

733 

991 

1259 

1543 

1823 

2113 

2399 

2711 

59 

257 

487 

739 

997 

1277 

1549 

1831 

2129 

2411 

2713 

61 

263 

491 

743 

1009 

1279 

1553 

1847 

2131 

2417 

2719 

67 

269 

499 

751 

1013 

1283 

1559 

1861 

2137 

2423 

2729 

71 

271 

503 

757 

1019 

1289 

1567 

1867 

2141 

2437 

2731 

73 

277 

509 

761 

1021 

1291 

1571 

1871 

2143 

2441 

2741 

79 

281 

521 

769 

1031 

1297 

1579 

1873 

2153 

2447 

2749  I 

83 

283 

523 

773 

1033 

1301 

1583 

1877 

2161 

2459 

2753 

89 

293 

541 

787 

1039 

1303 

1597 

1879 

2179 

2467 

2767 

97 

307 

547 

797 

1049 

1307 

1601 

1889 

2203 

2473 

2777 

101 

311 

557 

809 

1051 

1319 

1607 

1901 

2207 

2477 

2789 

103 

313 

563 

811 

1061 

1321 

1609 

1907 

2213 

2503 

2791 

107 

317 

569 

821 

1063 

1327 

1613 

1913 

2221 

2521 

2797 

109 

331 

571 

823 

1069 

1361 

1619 

1931 

2237 

2531 

2801 

113 

337 

577 

827 

1087 

1367 

1621 

1933 

2239 

2539 

2803 

127 

347 

587 

829 

1091 

1373 

1627 

1949 

2243 

2543 

28  f  9 

131 

359 

593 

839 

1093 

1381 

1637 

1951 

2251 

2549 

2833  i 

137 

353 

599 

853 

1097 

1399 

1657 

1973 

2267 

2551i2837| 

139 

359 

601 

857 

1103 

1409 

1663 

1979 

2269 

2557  2843  • 

149 

367 

607 

859 

1109 

1423 

1667 

1987 

2273 

2579  2851  i 

151' 

373 

613 

863 

1117 

1427 

1669| 

1993 

2281 

2591 

28571 

157' 

379 

617 

871 

1123 

1429 

16931 

1997 

2287 

2593 

2861J 

84  composite  numbers.  .    [chap.  17 

Resolution   of   Composite   Nuaibers   into   their  Primb 
Factors. 

1.  What  are  the  prime  factors  of  144  ? 

OPERATION.     Explanation. — Divide    the   144    by   any  prime 

2)144  number,  greater  than  a  unit,  that  is  contained  in  it 

without  a  remainder;  and  divide  this  quotient  in 

2)72  *^^  same  manner,  and  so  continue  dividing  until  the 

quotient  obtained  is  a  prime  number.    Then,  a  unit, 

2)36  *he  several  divisors,  and  the  last  quotient  will  be  the 

prime  factors  required.     Proceeding,  thus,  we  find 

2)18  the  prime  factors  of  144  to  be  1,  2,  2,  2,  2,  3,  and  3. 

3)9 
3  •^ 

2.  What  are  the  prime  factors  of  96  ? 

3.  What  are  the  prime  factors  of  360  ? 

4.  What  are  the  prime  factors  of  36  ? 

5.  What  are  the  prime  factors  of  56  ? 

6.  What  are  the  prime  factors  of  480  ? 
*l.  What  are  the  prime  factors  of  500  ? 

8.  What  are  the  prime  factors  of  840  ? 

9.  Resolve  460  into  its  prime  factors  ? 
10.  Resolve  680  into  its  prime  factors  ? 


Divisors  or  Measures  of  Numbers. 

Art.  87".  A  divisor  or  measure  of  any  number  is  a 
number  that  is  contained  in  it  an  exact  number  of  times, 
without  a  remainder. 

1.  What  are  the  divisors  of  *r2  ? 

Explanation We  first  find  the  prime  factors  of  72,  which 

are  1,  2,  2,  2,  3,  and  3.  A  number  is  evidently,  divisible  by 
its  prime  factors  and  the  products  arising  from  every  combina- 
tion of  them.  A  unit  and  the  factor  2  with  all  the  products 
arising  from  2,  2,  and  2,  gives  1,  2,  4,  and  8.  A  unit  and  the 
factor  3  with  all  the  products  arising  from  3  an(f  3,  gives  1,  3, 


ART.    88.]  COMMON   MEASURE.  85 

and  9.    The  various  products  arising  from  the  products  already 
obtained,  may  be  found  by  multiphcation.     Thus, 

1+2+4+8 

1  +  3+9 

1  +  2+4  +  8  +  3  +  6+12  +  24  +  9  +  18  +  36  +  72. 
Therefore,  the  divisors  of  72,  are  1,  2,  4,  8,  3,  6,  12,  24,  9,  18, 
36,  and  72. 

2.  What  are  the  divisors  of  48  ? 

3.  Find  all  the  divisors  of  96. 

4.  Find  all  the  divisors  of  144. 

5.  Find  all  the  divisors  of  360. 

Common  Measure  or  Divisor. 

Art.  88.  A  Common  measure  or  divisor  of  two  or 
more  numbers,  is  any  number  th*t  is  contained  in  each  of 
them  a  whole  number  of  times  without  a  remainder.  Thus, 
5  is  a  common  measure  of  10  and  15. 

1.  Fiud  all  the  common  measures,  or  divisors  of  144 
and  360. 

Explanation I  first  find  the  prime  fac- 
tors common  to  both  numbers,  which  I 
have  marked  by  *. — Since,  1,  2,  2,  2,  3, 
and  3  are  the  only  prime  factors  that  are 
common  to  144  and  360,  it  follows  that 
each  of  these  factors,  together  with  the 
products  arising  from  their  various  com- 
binations will  be  all  the  divisors  of  the 
two  numbers,  1.  2,  4,  and  8  are  all  the  di- 
visors arising  from  the  common  factors, 
2,  2,  and  2,  1,  3,  and  9  are  all  the  divisors 
arising  from  the  common  factors  3  and  3. 
The  divisors  arising  from  the  combinations 
of  the  above  divisors  are  found  by  multipli- 


OPERATION. 

*2)144          360 

*2) 

72 

180 

*2) 

36 

90 

«3) 

18 

45 

»3) 

6 

15 

cation, 

2              5 

.     Thus, 
1+2+4+8 
1+3+9 

1+2+4+8+3+6+12+24+9+18-4-36+72 
Hence ;  1,  2,  4,  8,  3,  6,  12,  24,  9,  18,  36,  and  72  are  all  the 
common  divisors  of  144  and  360. 


86  GREATEST   COMMON    MEASURE.  [cHAP.    IV. 

2.  Find  all  the  common  divisors  of  24  and  48. 

3.  Find  all  the  common  divisors  of  48,  96,  and  120. 

4.  Find  all  the  common  divisors  of  180,  360,  and  480. 

5.  Find  all  the  common  divisors  of  60,  120,  and  180. 

Greatest  Common  Measure. 

Art.  89.  The  greatest  common  measure,  or  the  great- 
est common  divisor  of  two  or  more  numbers,  is  the  greatest 
number  that  is  contained  in  each  of  them  a  whole  number 
of  times  without  a  remainder.  Thus,  7  is  the  greatest 
common  measure  of  35  and  42. 

1.  What  is  the  greatest  common  measure  of  126,  294, 
and  462  ? 

•  OPERATION. 

(  126=2*  X  3*  X  3   X7* 
The  prime  factors  of  \  294=2*x3*x7*x7 
(462=2*x3*x7*xll 

Explanation Since  a  number  is  divisible  only  by  its  prime 

factors  and  the  various  products  of  them,  it  follows  that  the 
product  of  all  the  factors  that  are  common  to  any  two  or  more 
numbers,  must  be  the  greatest  common  measure  of  these  num- 
bers. 

The  factors  marked  (^)  are  common  to  all  these  numbers, 
hence  their  product  is  the  greatest  common  measure,  or  divisor 
of  these  numbers;  which  is  2x3x7=42. 

2.  What  is  the  greatest  common  measure  of  462  and 
t70? 

3.  What  is  the  greatest  common  divisor  of  140,  105, 
And  245? 

4.  What  is  the  greatest  common  divisor  of  210,  350, 
and  770  ? 

5.  Wliat  is  the  greatest  common  measure  of  286,  429, 
and  715  ? 

Art.  90.  The  greatest  common  measure  of  two  or 
more  numbers  may  also  be  found  by 

Dividing  the  larger  number  by  Ike  smaller^  and  the  preceding 
divisor  ly  the  remainder,  (if  there  be  any,)  and  so  contiwiu 


ART.    90.]  PRACTICAL    QUESTIONS.  87 

to  divide  the  preceding  divisor  by  the  last  remainder  until 
nothing  remains,  then  will  the  last  divisor  he  the  greatest  com' 
mon  measure. 

Note— If  there  are  more  than  two  numbers  ;  first  find  the  greatest  common 
measure  of  two  of  them,  and  then  take  this  divisor  and  the  remaining  num- 
Der  and  proceed  as  before. 

6.  What  is  the  greatest  commoa  measure  of  105  and 
490? 

OPERATION.  Explanation. — If  the  remainder, 

105)490(4  (if  3.ny  after  division)  will  divide  the 

420  preceding  divisor,  it  will  also  divide 

the  dividend,  as  that  is  the  sum  of 

70)105(1  ^  certain  number  of  times  the  divi- 

70  sor  and  this   remainder :  it  is  also 

the  greatest  divisor  of  the  two  nuro- 

35)70(2      ^6rs,   as   that    divisor  is   the    same 
70  as  the  greatest  divisor  of  the  remain- 

der  and  the  preceding  divisor. 

0  This  is  rendered  plain  by  inspec- 

tion. Since  35  is  contained  in  70,  it 
is  contained  in  105,  (the  sum  of  70  and  35,)  also  in  490,  (the 
sum  of  70  und  4  times  105  :)  and  is  the  greatest  divisor  of  105 
and  490,  as  it  is  the  greatest  divisor  of  itself  and  70  the  largest 
numbers,  taken  at  pleasure,  that  will  produce  the  105  and  the 
490.     The  105=35-f  70  and  490=4x35+70. 

T.  What  is  the  greatest  common  measure  of  3094  and 
4420  } 

8.  What  is  the  greatest  common  measure  of  296  and  40t  ? 

9.  What  is  the  greatest  common  measure  of  360  and  480  ? 

10.  What  is  the  greatest  common  measure  of  268  and 
286? 


PRACTICAL    QUESTIONS    IN    COMMON   MEASURE. 

1.  A  farmer  has  120  bushels  of  wheat  and  460  bushels 
of  rye,  which  he  is  desirous  of  putting  into  boxes  of  equal 
size,  without  mixing  the  two  kinds  of  grain.  How  much 
will  the  largest  boxes  that  can  be  used  hold  .'' 

2.  A  had  $480;  B  $960;  and  C  $360,  which  they  were 
desirous  of  separating  into  different  parcels,  each  contain- 


88  MULTIPLES,  [chap;  IV. 

ing  the  same  number  of  dollars.     What  ib  the  greatest 
number  of  dollars  that  each  parcel  can  contain  ? 

3.  A  speculator  has  in  one  place  240  acres  of  land,  in 
another  480,  and  in  another  640,  and  -wishes  to  divide  the 
whole  into  fields  that  shall  be  of  equal  size,  and  contain- 
ing the  greatest  number  of  acres  circumstances  will  allow. 
What  will  be  the  number  of  acres  in  each  field  ? 

Multiples. 

Art,  91.  A  Multiple  of  any  number,  is  a  number 
that  will  contain  it  a  whole  number  of  times,  without  a 
remainder.     Thus,  21  is  a  multiple  of  3. 

Art,  92.  A  common  multiple  of  any  two  or  more 
numbers,  is  a  number  that  will,  when  divided  by  each  of 
them,  give  an  integer  for  a  quotient.  Thus,  24  is  a  com- 
mon multiple  of  4  and  8. 

Art.  93.  The  least  common  multiple  of  any  two  or 
more  numbers,  is  the  smallest  number  that  will,  when 
divided  by  each  of  them  give  an  integer  for  a  quotient. 
Thus  24  is  the  least  common  multiple  of  6,  8,  and  12. 

Art.  94.  To  find  the  least  common  multiple. 

Place  the  numbers  in  a  horizontal  line.  Then  divide  by 
any  prime  number  greater  than  a  unit  that  will  divide  the 
most  of  the  given  numbers  without  a  remainder,  and 
place  the  quotients  thus  obtained,  with  the  undivided 
numbers  in  a  liue  beneath;  thus  continue  to  divide  until 
no  number  greater  than  a  unit  will  divide  any  two  or  more 
of  them  without  a  remainder.  Then  the  product  of  all 
the  divisors,  the  last  quotients,  and  the  undivided  num- 
bers will  be  the  least  common  multiple. 

1.  What  is  the  least  common  multiple  of  6,  9,  and  30  ? 

OPERATION. 

2)6        9        30 

3)3        9        15 

13  5 

2  X  3  X  3  X  5  =  90  is  the  least  common  multiple. 


ART.    94.]  ABBREVIATED    OPERATIONS.  89 

Explanation. — Since  the  numbers  6,  9,  and  30  are  composed 
of  the  prime  factors  2,  3,  3,  and  5,  or  a  certain  number  of 
them,  it  follows  that  their  product  will  be  a  common  multiple 
of  these  numbers; — and  as  all  these  factors  are  necessary  to 
produce  the  above  numbers,  their  product  must  be  their  least 
common  multiple. 

2.  What  is  the  least  common  multiple  of  12, 16,  and  20  ? 

3.  What  is  the  least  common  multiple  of  15,  30,  and  9  ? 

4.  What  is  the  least  common  multiple  of  4,  8,  12,  16, 
and  20  ? 

5.  What  is  the  least  common  multiple  of  24,  48,  12  ? 

6.  What  is  the  least  common  multiple  of  18,  54,  2T, 
and  12  ? 

7.  What  is  the  least  common  multiple  of  12,  90,  and  45  ? 

8.  What  is  the  least  common  multiple  of  25,  45,  90 
and  5  ? 

9.  What  is  the  least  common  multiple  of  64,  8,  81,  24, 
and  12  ? 

10.  What  is  the  least  common  multiple  of  60,  120,  48, 
36,  24,  12,  6,  and  4  ? 


PRACTICAL    QUESTIONS    IN    COMMON    MULTIPLE. 

1.  What  is  the  smallest  sum  of  money  for  which  I  could 
purchase  a  number  of  hogs,  at  $9  each;  a  number  of 
cows,  at  $2t  each;  or  a  number  of  horses,  at  $60  each; — ■ 
and  how  many  of  each  could  I  purchase  for  that  sum  ? 

2.  What  is  the  smallest  number  of  bushels  of  corn  that 
will  fill  a  number  of  barrels,  each  containing  3  bushels;  a 
number  of  sacks,  each  containing  6  bushels;  or  a  number 
of  boxes,  each  containing  25  bushels  ? 

3.  If  one  team  can  haul  12  barrels  of  sugar,  at  a  load; 
another  15;  and  another  20; — what  is  the  smallest  num- 
ber of  barrels  that  will  make  a  number  of  full  loads  for 
any  of  three  teams  ? 


Abbreviated  Operations  in  Arithmetical  Calculations. 
There  are  many  abbreviated  methods  of  calculation,  in 


90  ABBREVIATED    OPERATIONS.  [cHaP.    IV 

particular  cases,  which  will  be  of  interest  to  the  student, 
and  of  much  importance  to  business  men.  We  will  men- 
tion a  few  of  them  to  awaken  a  habit  of  observation  on 
the  part  of  the  learner,  that  he  may  be  enabled  to  discover 
others  as  circumstances  may  require. 

Art.  95.  To  multiply  by  13,  14,  &lg.,  to  19. 

Write  the  product  of  the  unit's  figure  and  the  multipli- 
cand, under  the  multiplicand,  one  place  to  the  right  and 
then  add  them. 

Multiply  364^2  by  16. 

OPERATION. 

3G472  X  16 

218832 

583552 

Art.  96.  If  the  multiplier  is  a  unit  followed  by  one 
or  more  ciphers  and  a  significant  figure,  the  multiplicatioQ 
can  be  performed  by  writing  the  product  of  tiie  units' 
figure  and  the  multiplicand  as  many  places  to  the  right  of 
the  multiplicand  as  there  are  intervening  ciphers  -|-  1. 

Multiply  3642T3  by  104. 

OPERATION. 

364273     X  104 

1457092 


37884392  Ans. 
Multiply  8468327  by  10007. 

OPERATION. 

3468327    x  10007 

24278289 


34707548289  Ans. 


Art.  97.  .To  multiply  by  21,  31,  41,  &c.  to  91. 

Place  the  product  of  the  tenh  figure  and  the  multiplicand, 
under  the  multiplicand,  so  that  its  unit  figure  shall  be 
under  the  tens  of  the  multiplicand. 

Multiply  3476  by  41. 


ART.    99.]  ABBREVIATED    OPERATIONS.  91 


OPERATION. 

3476     X  41 
13904 


142516 

Should  there  be  ciphers  between  the  unit  and  the  other 
significant  figure  of  the  multiplier;  write  the  product  of 
the  significant  figure  one  more  place  towards  the  left  for 
every  cipher. 

Multiply  36U32  by  tOl. 

OPERATION. 

367432x701 
•2572024 


257569832  Ans. 
Multiply  46^321  by  80001. 

OPERATION. 

467321  X  80001 
3738568 


37386147321  Ans. 

Art.  98.  To  multiply  by  any  number  of  9's.  From 
the  multiplicand  with  as  many  ciphers  annexed  as  there 
are  9's  in  the  multiplier,  subtract  the  multiplicand. 

Multiply  34682  by  9999. 

OPERATION. 

346820000 
34682 


346785318  Ans. 

Explanation — 9^99=  10000 — 1,  consequently,  by  annexing 
four  ciphers  to  the  multiplicand,  we  have  taken  it  one  time 
more  than  we  should  have  done,  hence  by  subtracting  the 
multiplicand  gives  the  correct  result. 

Art.  99.  If  the  multiplier  is  an  Aliquct  Part  of 
any  number  of  tens,  hundreds,  or  thcAisands;  multiply  by 
the  number  of  tens^  hundreds,  or  thousands^  of  which  the  mul- 
tiplier is  an  aliquot  jpart,  then  take  the  same  jpart  of  the 
product  thus  found. 


ABBREVIATED    OPERATIONS.  CHAP.  IV 


Aliquot  Parts. 

12i  =  I  of  100 
16f  =  :^  of  100 
25  =  j  of  100 
50  =  I  of  100 
75  =  I  of  300 
331  =  t  of  100 
13^  =  i  of  40 


3   3 

&C.  &C. 


1.  Multiply  3248  by  12i. 

121  X  8  =  100,  therefore  12^  is  one-eighth  of  100. 

OPERATION. 

8)324800 

40600  Ans. 

2.  Multiply  86432  by  25. 

25  X  4  =  100,  therefore  25  is  one-fourth  of  100. 

operation. 
4)8643200 


2160800  Ans. 

3.  Multiply  846828  by  ^^. 

33^  X  3  =  100,  therefore  33^  is  one-third  of  100. 

OPERATION. 

3)84682800 
28227600  Ans. 

Art.  100.  Any  number  ending  in  5,  that  is  expressed 
by  two  figures,  can  be  squared  mentally. 

Tht  two  right  hand  figures  of  the  square  number  will 
always  he  25,  the  remaining  figures  on  the  left,  will  he  the 
product  of  the  digit  in  terHs  'place  and  a  figure  a  unit  greater. 

1.  What  is  the  square  of  25. 

25  X  25  =  625 

By  inspecting  the  following  multiplication,  the  reason 
of  this  method  of  squaring  a  number,  expressed  by  two 
figures,  that  ends  in  5,  will  become  evident.  This  meth- 
od of  squaring  a  quantity  will  apply  to  a  number  expressed 


IRT.    101.]  ABBREVIATED   OPERATIONS.  93 

by  three,  or  more,  figures ;  providing  the  figure  occupying 
the  unit's  place  is  5. 

OPERATION. 

25  =        20  +  5 
25  ==        20-1-5 

Remark — ^Commence  at 

5x20-f-25  the  right  and  multiply  by 

20x20-f-  5x20  each  figure  separately. 

'  20  x20-f-10x  20-1-25 
The  product  is  composed  of  (10  +  20)  times  20,  +  25, 
or  30  times  20  +  25  =  600  +  25  =  625. 
45  squared  =  2025 
85  squared  =  7225 
&c.  &c. 

*125  squared  =  15625. 
&c.  &c. 

Art.  101.  The  square  of  any  number  and  a  half  is 
equal  to  the  ^product  of  that  number  and  a  number  a  unii 
greater,  increased  by  one-fourth. 

9-J  squared  =  9xl0+^=90|- 

8i        «       =  8x  9+1=72^ 

&c.  &c. 

EXAMPLES   IN   ABBREVIATED   MULTIPLICATION. 

The  pupil  should  be  required  to  give  the  reason  of  all 
abbreviated  operations. 

1.  Multiply  46234  by  13. 

2.  Multiply  8647  by  16. 

3.  Multiply  84672  by  19. 

4.  Multiply  46732  by  103. 

5.  Multiply  68472  by  107. 

6.  Multiply  723246  by  1009. 

7.  Multiply  67234  by  21. 

8.  Multiply  846232  by  41. 

9.  Multiply  8467231  by  81. 

10.  Multiply  102324  by  701 

11.  Multiply  347234  by  6001. 

12.  Multiply  4726846  by  80001. 

*  Consider  the  12  on  the  left  of  the  5  as  one  number,  and  multiply  it  by  « 
number  a  unit  greater. 


94  ABBREVIATED    OPERATIONS.  [CHAP.    IV. 

13.  Multiply  4862321  by  12»-. 

14.  Multiply  846232  by  33^. 

15.  Multiply  *r23246  by  16|. 

16.  Multiply  8462342  by  25. 
lY.  Square  25  mentally. 

18.  Square  35,  45,  55,  65,  T5,  85,  aud  95,  mentally, 

19.  Square  125,  135,  145,  and  155,  mentally. 

20.  Square  ^,  5^,  6^  H,  8i,  9i,  10^,  11^,  and  12i, 
mentally. 

Art.  102.  Any  number  is  divisible  by  another  when  it 
contains  the  same  prime  factors  as  that  numbe? 

Hence,  to  divide  one  number  by  another,  resolve  them 
into  their  prime  factors  and  reject  equal  factors  from  each ; — 
the  product  of  the  remaining  factors  of  the  dividend  will 
be  the  quotient.  Should  the  divisor  not  be  a  measure  of 
the  dividend,  there  will  be  factors  remaining  in  the  divisor 
also.  In  such  cases,  the  product  of  the  remaining  factors 
in  the  dividend,  divided  by  the  product  of  the  remaining 
factors  in  the  divisor,  will  give  the  quotient. 

1.  Divide  1260  by  84. 

OPERATION. 


The  prime  factors  of  < 


1260  =  2X2X3X3X5X1. 
84      =2X2X3X1 


I^i^'^d^^^'  ^X^X3X3X  5x;^  ^  15        tient. 
Divisor,     ^  x  ^  X  $  X  ^ 

2.  Divide  3780  by  420. 

3.  Divide  6615  by  315. 

4.  Divide  46305  by  63. 

5.  Divide  2205  by  378. 

OPERATION. 

mi        '      e    A.        v^  2205  =  3X3X5X7X7. 
The  prime  factors  of  |  3^3    ^  3  ^  3  ><  3  ><  3  ^  t. 

Dividend,  $  X^  X  5  X  ^  X  7  ^  35  ^  ^^  ^^^ 
Divisor,     2  X  $  X  $  X^X^        6  ' 

Remark. — The  pupil,  from  what  has  been  said,  will  readily  discover  otk«r 
useful  lavthods  of  abbreviating  the  operations  of  the  Fundamenal  rules, 


ART.  108.]  FRACTIONS.  95 

Properties  of  NuiiBERS. 

Art.  103.  All  numbers  terminating  on  the  right  with  0, 
2,  4,  6,  or  8,  are  divisible  by  2 ;  since  each  of  the  numbera 
2,  4,  6,  8  and  10  contain  a  factor  2. 

Art.  104.  All  numbers  terminating  on  the  right  with  0, 
or  5,  are  divisable  by  5;  since  each  of  the  numbers,  5  and  10, 
contain  a  factor  5. 

Art.  105.  If  the  two  right-hand  figures  of  any  number 
are  dimnble  by  4,  the  whole  number  will  be  divisible  by  4. 
For,  if  there  is  any  remainder  next  preceding  the  two 
fi(i:ures  on  the  right,  it  will  be  a  certain  number  of  times 
100;  and  4  is  a  measure  of  100,  since  the  100  contains  the 
same  prime  factors  as  4 ;  and  as  it  is  also,  a  measure  of 
the  two  right-hand  figures;  it  is  a  measure  of  the  whole 
number. 

Art.  106.  If  the  three  right-hand  figures  of  any  number 
are  divisible  by  8,  the  whole  number  will  be  divisible  by  8. 
For  the  remainder,  (if  any,)  next  preceding  the  three  figures 
on  the  right,  will  be  a  certain  number  of  times  1000;  and 
8  is  the  measure  of  1000,  since  the  1000  contains  the  same 
prime  factors  as  8;  the  8  being  also,  a  measure  of  the  three 
right-hand  figures; — it  must  be  a  measure  of  the  whole 
number. 

Art.  lOT.  When  the  sum  of  the  digits  of  any  number 
is  divisible  by  3  or  9,  the  number  itself  is  divisible  bj!  3  or  9. 
(For  the  reason,  see  Art.  74.) 


CHAPTER  Y. 

FRACTIONS 


Abt.  1 08.  A  Fraction  is  an  expression  denoting  one 
or  more  of  the  equal  parts  into  which  a  unit,  or  any  col- 
lection of  units  may  be  divided. 

There  are  two  kinds  of  fractions  employed  in  Arithme- 


96  FRACTIONS.  [chap.    V. 

tical  calculations;  namely,  Common  Fractions  and  Decimal 
Fractions. 

The  Common  Fractions  have  generally  been  called  Vul- 
gar Fractions ;  the  word  vulgar,  meaning  common. 

Common  Fractions. 

Art.  109.  A  Common  Fraction  consists  of  two  num- 
bers, one  written  above  the  other,  with  a  short  horizontal 
line  between  them. 

The  number  above  the  line  is  called  the  Numerator^  and 
shows  how  many  of  these  parts  are  considered,  or  taken. 

The  number  below  the  line  is  called  the  Denominator, 
and  shows  into  how  many  equal  parts  the  unit  or  integer  is 
divided. 

I,  |,  4,  f ,  &c.,  are  Common  Fractions  and  are  read 

Numerator.  1     OuC 

Denominator.         3    Third  Of  One. 
Numerator.  2   TwO  >  ce^x,      e  n 

"  TTi"^!-      If  >  or  one  fiitn  of  2. 

Denominator.         5    Fifths  Of  OnC,  J 


or  one  seventh  of  4. 


Numerator.  4    FOUT 

Denominator.      7  Sevenths  of  onc. 

Numerator.  5     FivC  >  •    ^i       y.  r 

r  o-  XT,     e  /  or  one  sixth  of  5. 

Denominator.        6     Slxths  of  OUC,    J 

By  inspecting  the  above  expressions,  it  will  be  observed 
that  they  are  unperformed  operations  in  division.  The 
denominators  being  the  divisors,  and  the  numerators  the 
dividends.  Hence,  a  Common  Fraction  may  be  considered 
a  method  of  expressing  division.  (See  Remark  2,  Art,  45.) 

In  the  fraction  f ,  the  numerator,  5,  is  the  dividend,  and 
the  denominator,  9,  is  the  divisor 

Art.  no.  When  the  numerator  of  a  fraction  is  less 
than  the  denominator,  the  value  is  less  than  a  unit;  as,  f. 

2.  When  the  numerator  of  a  fraction  is  equal  to  the  de- 
nominator, the  value  is  a  unit;  as,  f  =  1. 

3.  When  the  numerator  of  a  fraction  is  greater  than 


ART.    112.]  COMMON    FRACTIONS.  91 

the  denominator,  tlie  value  is  greater  than  a  unit;   as, 

J=ii. 

Art.  111.  There  are  five  kinds  of  Common  Frac- 
tions, namely;  Proj^er,  Im'projper,  Simple,  Compound,  and 
Complex. 

A  Proper  Fraction  is  one,  the  numerator  of  which  is 
less  than  the  denominator;  therefore,  its  value  is  less  than 
a  unit;  as,  |,  f,  |,  &c. 

An  Improper  Fraction  is  a  fraction,  the  numerator  of 
which  is  equal  to,  or  greater  than  the  denominator;  there- 
fore, its  value  is  equal  to,  or  greater  than  a  unit;  as,  |-,  |, 
^,  &c. 

A  Simple  Fraction  is  one  in  which  the  numerator  and 
denominator  each  consist  of  an  integer  ;  and  may  be  either 
a  proper  or  an  improper  fraction.  * 

A  Compound  Fraction  is  a  fraction  of  a  fraction,  or  any 
number  of  fractions  connected  by  the  word  of;  as,  f  of  f 
of  I  of  |. 

A  Complex  Fraction  is  a  fraction  that  has  a  fraction, 
or  a  mixed  number,  in  the  numerator  or  denominator,  or 

in  both;  as  — ;  —  &c. 
4J     31 

A  Mixed  Number  consists  of  an  integer  and  a  fraction; 
as,  H.»  24f ,  &c. 

Art.  112.  The  Terms  of  a  fraction  are  two  in  num- 
ber, the  numerator  and  the  denominator. 

To  invert  a  fraction,  cause  the  numerator  and  the 
denominator  to  change  places.  Thus,  |  when  inverted, 
becomes  f . 

Any  whole  number  may  be  expressed  fractionally  by 
writing  a  unit  below  it  for  a  denominator.     Thus, 

4  =  f ,  and  is  read    4  ones,  or  four. 

7=  ^,     "     "     "       7  ones,  or  seven. 

9  =  f ,     "     "     "       9  ones,  or  nine. 

12  ==  V,     "    "    "    12  ones,  or  twelve. 


98  ,  FRACTI0N3.  [cHAP  ▼. 

Reduction  of  Common  Fractions. 

Art.  113.  Reduction  of  Fractions  is  changing  them 
from  one  form  to  another  while  their  value  remains  the 
1  same. 

Art.  114.  Reduction  of  Mixed  Numbers  to  Improper 
Fractions. 

1.  In  25|  how  many  thirds  ? 

Solution. — In  1  there  are  f ,  and  in  25  there  are  25  times 
%=\^,  which  added  to  |=  V,  consequently,  25|=y. 

2.  In  43f ,  how  many  fourth  ? 

3.  In  14:6f ,  how  many  sevenths  ? 

4.  In  2361,  how  many  halves  ? 

5.  Reduce  684|  to  an  improper  fraction. 

6.  Reduce  783|-  to  an  improper  fraction. 

1.  Reduce  1862y'^  to  an  improper  fraction. 

8.  Reduce  2864y\  to  an  improper  fraction. 

9.  Reduce  86232^^  to  an  improper  fraction. 

10.  Reduce  76432yy8  to  an  improper  fraction.     ♦ 

Art.  1 15.  Reduction  of  Improper  Fractions  to  Mixed 

Numbers. 

1.  Reduce  ^^^  to  a  mixed  number. 

Solution. — In  one  there  are  f ;  therefore,  1  third  of  the 
'number  of  thirds,  equals  the  number  of  whole  ones.  1 
third  of  24t=82i.  Hence,  2|i=82i. 

2.  Reduce  -^-f^  to  a  mixed  number. 

3.  Reduce  ^f  ^  to  a  mixed  number. 

4.  Reduce  ^|^  to  a  mixed  number. 
6.  Reduce  -^-f^  to  a  mixed  number. 
6.  Reduce  -V^-^  to  a  mixed  number. 
1.  Reduce  -i-V^  to  a  mixed  number. 

8.  Reduce  -f  f-^  to  a  mixed  number. 

9.  Reduce  -V/-^  to  a  mixed  number. 
10.  Reduce  a.ii|AX  to  a  mixed  number. 


ART.  116.]  PROPOSITIONS.  99 

Propositions. 

Art.  116,  Proposition!. — Multiplying  the  numerator 
of  a  fraction  hy  any  number,  the  denominator  remaining  the 
same,  multiplies  the  value  of  the  fraction  by  that  number. 

The  denominator  of  a  fraction  shows  into  how  many  equal  parts 
the  quantity  is  divided,  and  therefore,  designates  the  size  of  the 
parts  compared  with  that  quantity.  The  numerator  shows  how 
many  of  these  parts  are  taken  ;  hence,  multiplying  the  numerator 
increases  the  value  of  the  fraction  as  many  times  as  there  are  units 
in  the  multiplier,  if  the  denominator,  that  is,  the  size  of  the  parts 
remains  the  same. 

Proposition  2. — Dividing  the  denominator  of  a  fraction 
by  any  number,  the  numerator  remaining  the  same,  multiplies 
the.  value  of  the  fraction  by  that  member . 

The  numerator  of  a  fraction  shows  how  many  parts  are  taken, 
and  the  denominator  measures  the  size  of  these  parts,  compared 
with  the  quantity  referred  to ;  if  we  divide  the  denominator  by 
any  number  it  diminishes  the  number  of  parts  into  which  the  thing 
is  divided,  and,  therefore,  increases  their  size  proportionally  • 
hence,  the  value  of  the  fraction  is  multiplied  by  the  same  number, 
if  the  numerator,  that  is,  the  number  of  parts  taken,  remains  the 
same. 

Proposition  3. — Multiplying  the  denominator  of  a  frac- 
tion by  any  number,  the  numerator  remaining  the  same,  divides 
the  value  of  the  fraction  by  that  number. 

The  numerator  of  a  fraction  shows  how  many  parts  are  taken, 
and  the  denominator  shows  into  many  equal  parts  the  unit  or 
thing  is  divided,  and  therefore,  designates  the  size  of  these  parts 
compared  with  the  unit  or  thing  divided.  Multiplying  the  denom- 
inator by  any  number,  increases  the  number  of  parts  into  whitch 
the  thing  is  divided,  as  many  times  as  there  are  units  in  the  mul- 
tiplier, and  necessarily  diminishes  their  size  proportionally  ;  there- 
fore, the  value  of  the  fraction  is  divided,  if  the  numerator  remains 
unchanged. 

Proposition  4. — Dividing  the  numerator  of  a  fraction  by 
'any  number,  the  denomi7iator  remaining  the  same,  divides  the 
value  of  the  fraction  by  that  number. 

Since,  the  denominator  shows  into  how  many  equal  parts  the 
unit  or  thing  is  divided,  and  the  numerator  shows  how  many  of 
these  parts  are  taken ;  it  follows,  that  dividing  the  numerator. 


100  FRACTIONS.  '^      [chap     IV. 

divides  the  value  of  the  fraction,  as  it  diminishes  the  number  of 
parts  taken  while  their  size  remains  the  same. 

Remark. — By  inspecting  proposition  1  and  3,  we  deduce 

Proposition  5. — Multiplying  the  numerator  and  denorri' 
inator  of  any  fraction  by  the  same  number,  does  not  change 
the  value  of  the  fraction. 

Remark. — By,  inspecting  proposition  2  and  4,  we  deduce 

Proposition  6. — Dividing  both  numerator  and  denomina- 
tor of  any  fraction  by  the  same  number,  does  not  change  the 
value  of  the  fraction. 

Multiplication  of  Fractions  by  Integers. 

Art.  IIT,  According  to  propositions  1st  and  2d,  to 

to  multiply  a  fraction  by  any  integer ;  Multiply  the  numera- 
tor of  the  fraction  by  that  number, — or  divide  its  de7iominator 
by  the  same  number. 

1.  If  1  bushel  of  apples"  cost  $4,  what  will  15  bushels 
cost? 

Solution. — If  1  bushel  cost  $4>  1^  bushels  will  cost 
15  X  $4  =  V  =  $8f 

2.  If  1  barrel  of  sugar  cost  $12|-,  (equal  to  %^^,)  what 
will  3  barpels  cost  ? 

Solution. — If  1  barrel  cost  %^\^,  3  barrels  will  cost  3 
times  $113.  =  112  _  1371, 

3.  What  cost  25  bushels  of  peaches,  at  $f  a  bushel } 

4.  What  cost  18  yards  of  broadcloth,  at  $6|  a  yard  ? 
6.  What  cost  4t  barrels  of  flour,  at  $5|  a  barrel  ? 

6.  What  cost  15  cows,  at  25f  each  ? 
1.  What  cost  52  barrels  of  cider,  at  $11  a  barrel  ? 
8.  What  cost  IT  hogsheads  of  molasses,  at  %i1^  a  hogs- 
head ? 

Division  of  Fractions  by  Integers. 

Art.  118.  According  to  Propositions  3d  and  4th,  to 

divide  a  fraction  by  any  integer ;  Divide  the  nu7nerator  of 
the  fraction  by  that  number, — or  muitiply  the  denominator  by 
the  same  number. 


ART,    119.]        DIVISION  OF  FRACTIONS  BY  INTEGERS.  101 

1.  If  16  yards  of  cloth  cost  $3ti,  what  will  1  yard  cost  ? 

2.  If  13  yards  of  ribbon  cost  32^  cents,  how  much  is 
that  a  yard  ? 

3.  If  8  books  cost  $13|,  how  much  is  that  a  piece  ? 

4.  If  14  lbs.  of  sugar  cost  93f  cents,  how  much  is  that 
a  pound  ? 

5.  If  It  barrels  of  sugar  cost  $282^,  how  much  is  that 
a  barrel  ? 

6.  What  cost  1  horse,  if  19  horses  cost  $t824f  ? 
1.  What  cost  5  oranges,  if  15  cost  35|^  cents  ? 

8.  What  cost  6  acres  of  land  if  It   acres  cost  $403f  ? 

9.  What   cost   3   bushels   of   flax-seed,   if    7   bushels 
cost  $2tf  ? 

10.  What  cost  4  horses  if  12  cost  $2104^  ? 

Art.  119.  According  to  proposition  6th,  to  reduce  a 
fraction  to  its  lowest  terms: 

Divide  both  numerator  and  denominator  by  any  number 
greater  than  a  unit,  that  is  contairied  in  them  both  without  a 
remainder  ;  proceed  in  the  same  way  with  the  successive  results, 
until  the  operation  can  be  carried  no  farther.  (See  Arti- 
cles 103  to  107,  Properties  of  numbers.) 
Or, 

Find  the  greatest  common  measure  of  the  numerator  and 
denominator  (by  Art.  89  or  Art.  90)  and  divide  them  by  it. 
Or, 

Resolve  both  numerator  and  denominator  into  their  prime 
factors  and  reject  equal  factors  from  each;   (See  Art.  102) 
the  result  will  be  the  fraction  reduced  to  its  lowest  terms. 
1.  Reduce  |f^  to  its  lowest  terms. 

Operation  by  the  last  method. 


The  prime  factors  of    \  ^— 
^  I  600= 


2.  Reduce  f  f  f  to  its  lowest  terms. 

3.  Reduce  -j-f  ^  to  its  lowest  terms. 

4.  Reduce  -f  f  to  its  lowest  terms. 

5.  Reduce  f  ||  to  its  lowest  terms. 

6.  Reduce  |;^|  to  its  lowest  terms. 


102  FRACTIONS.  [chap.   V. 

*l.  Reduce  j%W  to  its  lowest  terms, 

8.  Reduce  ||f  to  its  lowest  terms. 

9.  Reduce  |m  to  its  lowest  terms. 
10.  Reduce  {UH  ^o  its  lowest  terms. 


Art.    120.    Reducton   of   Compound  Fractions    to 
Simple  ones. 

Remark. — The  word  of  in  the  following  questions,  is  equivalent  to  the 
sign  of  multiplication;  therefore,  in  its  stead  the  sign  X,  may  be  used. 

1.  Reduce  |  of  f  to  a  simple  fraction. 

OPERATION.  Solution.— i  of  ^  is  yV  and  ^  of  |  is  4 

2    4     8    A  *i°^6s  yV,  which  is  y\ ;  and  if  ^  of  f  is  y\, 

oX^=^R  ^^^        f  of  !  is  twice  y^^,  which  are  yV     There- 
fore, f  off  =yV 

Remark. — From  the  above  solution  we  observe,  that  to  reduce  a  compound 
fraction  to  a  simple  one,  we  multiply  all  themimerators  together  for  a  new  numera 
tor,  and  all  the  denominators  for  a  neto  denominator. 

2.  Reduce  |  of  |  to  a  simple  frastion. 

3.  Reduce  |  of  y\  to  a  simple  fraction. 

4.  Reduce  |  of  |  of  y^^-  to  a  simple  fraction. 

5.  Reduce  ^  of  i  of  |  to  a  simple  fraction. 

6.  Reduce  ^  of  |  of  f  to  a  simple  fraction. 


Cancellation. 

Art.  121.  To  reduce  a  Compound  Fraction  to  a 
Simple  one  by  Cancellation. 

Write  the  fraction  down  with  the  sign  of  multiplication 
between  them,  and  cancel  or  reject  all  the  factors  that  are 
common  to  the  numerators  and  deTwminators,  (which  by  Pro- 
position 6th,  under  Art.  116,  does  not  change  the  value  of  the 
fraction  ;)  then  multiply  the  remaining  numerators  together 
for  a  n£w  numerator,  and  the  remairdng  denominators  for 
a  new  denominator. 

Take  for  illustration  the  6th  example. 
1      I     ^_}_ 


ART.    122.]  CANCELLATION.  103 

Explanation.— First  cancel  the  3  of  the  numerator  against 
the  3  of  the  denominator,  by  drawing  a  line  across  them  ]  then 
cancel  the  4  of  the  numerator  against  the  4  of  the  denomina- 
tor in  the  same  manner.  As  there  are  no  more  factors  com- 
mon to  both  numerator  and  denominator, — multiply  the  remain- 
ing numerators  together  for  a  new  numerator;  and  the  remain- 
ing denominators,  for  a  new  denominator. 

Art.  122.  If  any  numerator  and  denominator  have  a 
common  divisor,  divide  them  hoth  by  this  divisor,  and  use  ihi 
quotients  as  a  neiv  fraction. 

6.  Reduce  4  of  j\  of  |f  to  a  simple  fraction. 

OPERATION. 

*1  2 

8        3 

Explanation. — 5  being  a  divisor  of  the  5  in  the  numerator 
and  the  15  in  the  denominator,  we  divide  them  both  by  5  and 
cancel  the  5  and  the  15,  and  consider  the  quotients,  1  and  3, 
arising  from  this  division,  instead  of  the  5  and  15.  Next  can- 
cel the  7  in  the  numerator  and  the  7  in  the  denominator.  We 
observe  that  4  is  a  common  measure  of  the  8  in  the  numerator, 
and  of  the  12  in  the  denominator ;  therefore,  we  divide  by  it, 
and  cancel  the  8  and  12,  and  place  the  quotients  in  their  pro- 
per places.  As  there  are  no  more  factors  common  to  both 
numerator  and  denominator,  nor  any  number  that  will  divide 
them  both  without  a  remainder,  we  multiply  all  the  remaining 
numerators  together  for  a  new  numerator,  and  all  the  remain- 
ing denominators  for  a  new  denominator,  and  obtain  for  the 
answer  f . 

T.  Reduce  f  of  |-  of  j\  of  j\  to  its  simplest  form. 

8.  Reduce  4  of  |  of  |f  of  |  to  its  simplest  form. 

9.  Reduce  |  of  |  of  If  of  f  of  V  of  /,  to  its  simplest 
form. 

10.  Reduce  |  of  f  of  |  of  j%  of  f  of  ff  to  its  simplest 
form. 

11.  What  is  the  value  of  the  compound  fraction,  |  of  ^ 
ofiofyVofHof^V? 

*  In  practice  we  do  not  write  the  quotient  when  it  is  a  unit. 


104  FRACTIONS.  [chap.  V 

12.  What  is  the  value  of  the  compound  fraction  4  of 

2J    of  12.  of  i^  of  25   of  15  ? 
6      "^    3  0    "^    4  5    "^    S  3    "^    2  5    • 

Remark.— All  whole  and  mixed  numoers  that  oecur  in  compound  fractions, 
must  be  changed  to  improper  fractions  before  the  required  reduction  is  per- 
formed, 

13.  Reduce  ly\  of  3  of  y%  of  2f  to  its  simplest  form. 

14.  Reduce  yVj  of  3^  of  ly»j  of  l^  to  its  simplest  form. 

15.  Reduce  j\  of  4^  of  3^  of  /g  of  ^  of  li  of  3^  to  its 
simplest  form. 

16.  Reduce  |  of  9i  of  3^  of  |f  of  "^f  to  its  simplest  form. 
It.  Reduce  j\  of  3y\  of  y^  of  j\  of  3^  to  its  simplest 

form. 

18.  Reduce  yV  of  1}  of  /^  of  12f  of  y\  to  its.  simplest 
form. 

19.  Reduce  j\\  of  3"^  of  4  of  if  of  ^  to  its  simplest 
form. 

A  Common  Denominator. 

Art.  123.  Two  or  more  fractions  have  a  Common 
Denominator,  when  they  have  the  same  number  for  a 
denominator. 

1.  Reduce  f  and  |  to  equivalent  fractions  having  a  com- 
mon denominator. 

OPERATION. 

3  4_15  16  ,,  _  15,  16 
^'  "^      20'  20  20 

Explanation. — I  first  multiply  the  denominators  together  for 
a  common  denominator, — 4  times  5  are  20,  the  common  denom- 
inator. Since,  I  have  multiplied  the  denominator  4,  of  the 
fraction  f  by  5,  to  preserve  the  value  of  the  fraction,  I  multiply 
the  numerator  3,  by  the  same  number.  5  times  3  are  15  ; 
therefore,  f  equals  -^-f.  I  have  multiplied  the  denominator  5, 
of  the  fraction  |  by  4,  and  to  preserve  the  value  of  the  fraction, 
1  multiply  the  numerator  4,  by  the  same  number.  4  times  4  are 
16^  therefore,  f  equals  ^f . 

From  the  above  explanation,  to  reduce  fractions  to  equivalent 
ones  have  a  common  denominator,  we  infer  that  we  should 
Multiplu  all  the  denominators  together  for  a  common  denominator .^ 
and  each  numerator  by  all  the  denominators  except  its  own,  for  a 
new  numerator. 


ART.  124.]      THE   LEAST   COMMON   DENOMINATOR.  105 

Remark.— It  is  readily  observed  that,  by  the  above  process,  both  numerator 
and  denominator  of  each  fraction  is  multiplied  by  the  same  number,  which  by 
Proposition  5,  under  Art.  116.  does  not  change  the  value  of  the  fraction. 

2.  Reduce  f  and  |  to  equivalent  fractions  having  a 
common  denominator. 

3.  Reduce  4  and  ^  to  equivalent  fractions  having  a 
common  denominator. 

4.  Reduce  f  and  f  to  equivalent  fractions  having  a 
common  denominator. 

5.  Reduce  -f  and  ^  to  equivalent  fractions  having  a 
common  denominator.  _ 

6.  Reduce  -,%  and  y\  to  equivalent  fractions  having  a 
common  denominator. 

1.  Reduce  |,  |,  and  f  to  equivalent  fractions  having 
a  common  denominator. 

8.  Reduce  |,  f ,  and  |  to  equivalent  fractions  having 
a  common  denominator. 

9.  Reduce  |,  f  and  f  to  equivalent  fractions  having 
a  common  denominator. 

10.  Reduce  i,  i,  i,  and  f  to  equivalent  fractions  having 
a  common  denominator. 

The  Least  Common  Denominator. 

Art.  124.  The  least  common  denominator  of  two  or 
more  fractions,  is  the  least  common  multiple  of  their  denom- 
inators. Hence,  to  find  the  least  common  denominators 
of  two  or  more  fractions,  reduce  compound  fractions  to 
simple  ones,  whole  and  mixed  numbers,  to  improper  fr actions , 
and  all  to  their  lowest  terms  ;  then  find  the  least  common  mul- 
tiple of  the,  denominators  of  the  fractions,  (hy  Art.  9 4, J  and 
it  will  he  their  least  common  denominator. 

1 .  Reduce  y5_,  f ,  and  y\,  to  equivalent  fractions  having 
the  least  denominator. 

OPERATION. 

i,  i,  1  =  L^  so  y,  or  write  them  this,  1^?^^ 
2)12  6    18       36'  36  36  36 

3)6    3    9 

2    13 

2  X  3  X  2  X  3  =  36,  the  least  common  denominator. 

6* 


106  FRACTIONS.  [chap.  V. 

Solution. — The  remaining  part  of  the  work  is  to  reduce  each 
of  the  given  fractions  to  thirty  sixths,  without  changing  their 
value.  This  can  be  done  by  multiplying  the  terms  of  each 
fraction  by  a  number  that  will  cause  its  denominator  to  become, 
3G.  (See  Art.  116,  Proposition  5.)  To  find  what  number 
I  must  multiply  12  by  to  produce  36,  I  divide  the  36  by  12,  and 
find  it  to  be  3.  Multiply  both  numerator  and  denominator  of 
f3  by  3,  gives  if  =  -^\.  Proceed  in  the  same  way  with  the 
remaining  fractions. 

2.  Reduce  f,  y\  and  /^  to  equivalent  fractions  having 
the  least  common  denominator. 

3.  Reduce  |,  |,  and  y\  to  equivalent  fractions  having 
the  least  common  denominator. 

4.  Reduce  y^g,  g^,  and  -/p  to  equivalent  fractions  having 
the  least  common  denominator. 

5.  Reduce  -f,  9y\,  and  |f  to  equivalent  fractions  hav- 
ing the  least  common  denomhlator. 

6.  Reduce  y'V,  2|,  and  Sy^^  to  equivalent  fractions  hav- 
ing the  least  common  denominator. 

7.  Reduce  2|,  3y\,  and  3  -^^  to  equivalent  fractions 
having  the  least  common  denominator. 

8.  Reduce  f  of  |,  ^  of  y\,  and  |  of  |  to  equivalent 
fractions  having  the  least  common  denominator. 

9.  Reduce  i  of  f  of  |,  f  of  |,  |  of  f ,  and  f  of  f  to 
equivalent  fractions  having  the  least  common  denominator. 

10.  Reduce  2f  of  f ,  y^^  of  f ,  3|  of  ^^  of  f  and  i  of  f 
of  I  to  equivalent  fractions  having  the  least  common 
denominator. 


Addition  of  Common  Fractions. 

Art.  125.  Addition  of  common  fractions  is  the  process 
of  finding  the  sum  of  two  or  more  fractions. 

1.  What  is  the  sum  \,  f ,  |  and  |  ? 

OPERATION. 

1  +  f +1  +f=V,or2|.  Ans. 

2.  What  is  the  sum  of  f ,  f ,  f ,  V»  ^nd  |  ? 

3.  What  is  the  sum  of  ^,  f,  f ,  V,  and  f  ? 


ART.   126.]       SUBTRACTION  OF  COMMON  FRACTIONS.  107 

4.  What  is  the  sum  of  j\,  y\,  \\,  |f ,  |f ,  and  |4  ? 

5.  What  is  th^^  sum  of  If,  If,  ff,  H,  H,  H,  and  ||  ? 

Remark. — Reduce  con' pound  fractions  to  simple  ones,  and  mixed  numbers  to 
improper  fractions,  and  o/i  to  their  lowest  terms.  Also,  reduce  fractions  that 
have  diderent  denominatois  to  equivalent  ones  having  the  least  common  denom- 
inator. 

6.  What  is  the  sum  of  f ,  f  and  j\  ? 

OPERATION. 

3       7        7        18  4-21+14_53_o5     Ans 

2)2,      4,       6 
1;      2,       3 
2x2x2x3  =  24,  the  least  comoion  denominator.  . 

7.  What  is  the  sum  of  |-,  f ,  and  f'2  ? 

8.  What  is  the  sum  of  |,  4f ,  and  2^  ? 

9.  What  is  the  sum  of  4i,  8|,  and  8f  ? 

10.  What  is  the  sum  of  f  of  4  of  f ,  and  |  6f  f  ? 

11.  What  is  the  sum  of  |  of  W,  f  of  \%  \  of  f ,  and  f  ? 

12.  What  is  the  sum  of  ^  of  5i,  6^*3-  off  and  ^  of  /^  ? 
18.  What  is  the  sum  of  i,  i,  i,  f ,  1,  and  f  ? 

14.  What  is  the  sum  of  i,  |,  f,  f ,  I,  f ,  f,  and  |  ? 

Remark. — When  two  fractions  are  to  be  added,  the  numerator  of  each  being 
a  unit,  it  may  be  done  mentally,  by  taking  the  sura  of  the  denominators  for  a 
new  numerator  and  their  product  for  a  denominator. 

Thus,  the  sum  of  1  and  i  —  I-±4  =  ^ 
^         '       7x5       35 

15.  What  is  the  sum  of  ^  and  ^  ? 

16.  What  is  the  sum  of  J-  and  ^  ? 
It.  What' is  the  sum  of  \  and  \  ? 

18.  What  is  the  sum  of  |  and  \  ? 

19.  What  is  the  sum  of  ^  and  |  ? 

Subtraction  of  Common  Fractions. 

Art.  126.     Subtraction   of  common   fractions  is   the 
method  of  finding  the  difference  between  two  fractions. 
1.  From  I  subtract  3f . 


108  ,  FRACTIONS.  [chap.    V. 

OPERATION. 

I  -  f  =  f .  Ans. 

2.  From  f  take  f . 

3.  Fromf  take  f  * 

4.  From  |f  take  y\. 

5.  From  jf  take  j\. 

I    Remark. — Reduce  compound  fractions  to  simple  ones,  and  mixed  numher  to  t»». 

'  proper  fractions,  and  all  to  their  lowest  terms.  Also,  reduce  fractions  that 
have  different  denominations  to  equivalent  ones  having  the  least  common  de- 
nominator. 


6.  From  f  take  /j- 

OPERATION. 

=i      9i_in     n 

Ans, 


OPERATION. 

7      5  _21— 10_11 
2)8     12""     24     ""24 


2)4      6 

2      3 
2x2x2x3x=24,  tlie  least  common  denominator. 

1.  From  I  take  f . 

8.  From  f  take  |. 

9.  From  1^  take  |. 

10.  From  4i  take  1|. 

11.  From  8|  take  6f . 

12.  From  ^f  take  6f 

13.  From  |  of  f  take  }  of  |. 

14.  From  f  of  y^.  take  f  of  4  of  1. 

15.  From  |  of  1^  of  4  f  of  U  of  yV- 

16.  From  9yV  of  4}  take  i  of  |  of  f  of  4^. 

Remark. — When  both  the  fractions  have  a  unit  for  their  numerator,  the 
subtraction  may  be  performed  mentally  by  placing  the  product  of  the  denom- 
inators  under  their  diflference. 


Thus,   i  — i  = 

8  —  53 

•    4  0     ="f  JT' 

11.  From  1  take  i. 

18.  From  i  take  i. 

19.  From  i  take  yV- 

20.  From  I  take  yV- 

21.  From  4  take  -jV. 

ART.    128.]    MULTIPLICATION   OF   COM1|ON   FRACTIONS.  109 

22.  From  j\  take  Jg- 

23.  From  j\  take  j\. 


Multiplication  of  Common  Fractions. 

Art.  127".  Multiplication  of  common  fractions  is  the 
method  of  finding  the  product  of  two  or  more  fractions, 
or  of  integers  and  fractions. 

Art.  128.  To  multiply  one  fraction  by  another,  or  an 
integer  by  a  fraction.     First: 

Reduce  compound  fractions  to  simple  ones,   and  whole  or 
mixed  numbers  to  improper  fractions.     Then  proceed  as  in 
the  reduction  of  compound  fractions.     (See  Art.  121.) 
1.  Multiply  f ,  4,  If,  If,  and  ||,  together. 


OPERATION   BY   CANCELLATION. 

2       $       ^ 
^    $    X^    X^    $$     2 
-X-X— X— X— =-  Ans. 

d  ^  x$  x$   t$   s 

2.  Multiply  I  by  |. 

3.  Multiply  f  by  ||. 

4.  Multiply  I  by  ||. 

5.  Multiply  ^  by  41. 

6.  Multiply  together  f ,  |,  |,  and  -f. 

t.  Multiply  together  f  |,  |,  |f ,  and  |. 
8.  Multiply  together  21,  ||,  Si  and  jf 
9. ^Multiply  together  4i,  yV,  5^,  8^,  and  /j. 
10.  Multiply  together  |,  9i,  7|,  yV,  If,  and  H. 


PRACTICAL   QUESTIONS    IN    MULTIPLICATION    OF    FRACTIONS. 

Remark. — In  business  transactions  it  is  customary  to  add  1  cent  when  the 
fraction  is  equal  to  or  greater  than  a  half  of  a  cent,  and  to  omit  it  when  it  la 
less  than  the  half  of  a  cent. 


110  FRACTIONS.  [chap.    V. 

1.  What  cost  42  bushels  of  apples,  at  63f  cents  a 
bushel  ? 

2.  What  cost  Tf  dozens  of  eggs,  at  12^  cts.  a  dozen  ? 

3.  What  cost  13f  bushels  of  turnips,  at  3t^  cents  a 
bushel  ? 

4.  What  cost  lOf  yards  of  calico,  at  15i  cents  a  yard  ? 

5.  What  cost  t5^  pounds  of  sugar,  at  If  cts.  apouni? 

6.  What  cost  Sf^tons  of  hay,  at  $12f  a  ton  ? 

T.  What  cost   6|   bushels  of  apples,   at  31^  cents  a 
bushel  ? 

8.  What  cost  13|-  pounds  of  fish,  at  9f  cts.  a  pound  ? 

9.  What  cost   lt5  pounds  wool,  at  39f  cts.  a  pound  ? 

10.  What  cost  18f  yards  of  ribbon,  at  23^  cents  a  yard  ? 

11.  What  cost  18  pocket  handkerchiefs,  at  f  of  a  dollar 
each  ? 

12.  What  cost  22|  yards  of  selicia,  at  81|  cents  a  yard  ? 

13.  What  cost  35|  pounds  of  raisins,  at  18|  cents  a 
pound  ? 

14.  What  cost  t5f  bushels  of  wheat,  at  $1|  a  bushel  ? 

15.  What  cost  23|  cords  of  wood,  at  $3f  a  cord  ? 

16.  What^cost  212|  pounds  of  beef,  at  7^  cents  a  pound  ? 
It.  What'cost  14f  barrels  of  vinegar,  at  $1  Of  a  barrel  ? 

18.  What  cost  22f  barrels  of  sugar,  at  $15f  a  barrel  ? 

19.  What  cost  35i  tons  of  coal,  at  $9f  a  ton  ? 


Division  of  Common  Fractions. 

Art.  129.  Division  of  common  fractions  is  the  method 
of  dividing  one  fraction  by  another,  or  whole  numbers  and 
fractions  by  each  other. 

Remark. — When  the  fractions  have  a  common  denominator,  division  can 
be  performed  by  dividing  the  numerator  of  the  one  by  the  numerator  of  the 
other. 

1.  Divide  f  by  f . 

OPERATION. 


2.  Divide  |f  by  /y. 

3.  Divide  ^j  by  y\. 


ART.    130.]  PRACTICAL   QUESTIONS.  Ill 

4.  Divide  j^  by  y\. 


5.  Divide  ^^  by  ^\. 

6.  Divide  |  by  f . 

Solution. — 1  is  contained  in  |,  |  times ;  and  if  1  is  con- 
tained in  I,  I  times,  J-  is  contained  in  |,  4  X  |  times;  and 
f  i^  contained  in  it  i  of  4  X  |  times  =  |  x  |  =  f  times. 

Art,  130.  Hence,  to  divide  a  fraction  by  a  fraction, 
or  fractions  and  whole  numbers,  by  each  other,  we  merely, 
Invert  the  divisor  and  ^proceed  as  in  multiplication,  after  hav- 
ing reduced  comjpound  fractions  to  simple  ones,  and  whole  and 
mixed  numbers  to  improper  fractions. 

7.  Divide  f  by  f. 

8.  Divide  f  by  f . 

'      9.  Divide  21  by  4. 

10.  Divide  31  by  li. 

11.  Divide  H  by  6i 

12.  Divide  8^  by  6|. 

13.  Divide  1  of  f  by  2f . 

14.  Divide  f  of  |  by  2^. 

15.  Divide  3^  of  |  by  |  of  1^. 

16.  Divide  4^  times  3|  by  8|. 

17.  Divide  f  of  -J-f  by  -f  of  if. 

18.  Divide  2%  of  H  of  8f  by  4i  times  6f . 

19.  Divide  34  of  81  by  61  of  3i: 

20.  Divide  yV  of  |i  of  81  times  4  by  f  of  4^^. 

PRACTICAL    QUESTIONS    IN    DIVISION    OF    FRACTIONS. 

1.  At  $1  a  bushel,  how  many  bushels  of  apples  can  be 
bought  for  $20  ? 

2.  At  f  of  a  cent  a  piece,  how  many  oranges  can  be 
bought  for  14|  cents  ? 

3.  If  I  pay  4 1  cents  for  riding  1  mile,  how  many  miles 
can  I  ride  for  280  cents  ? 

4.  A  butcher  expended  $25t|^  for  sheep,  at  $lf  a  head; 
how  many  sheep  did  he  buy  } 


112  FRACTIONS.  [chap    V. 

5.  How  many  pounds  of  tea,  at  $lf  a  pound,  can  be 
obtained  for  $19f  ? 

6.  A  lady  bought  3^f  yards  of  calico  for  561  cents; 
how  much  did  it  cost  a  yard  ? 

7.  A  merchant  bought  96  sheep  for  $99|i;  how  much 
did  he  give  a  head  ? 

8.  How  many  tons  of  coal,  at  $8f  a  ton,  can  be  bought 
for  $97  ? 

9.  A  man  paid  $565^  for  a  farm,  giving  $21f  an  acre; 
of  how  many  acres  did  the  farm  consist  ? 

10.  At  $1^  a  day,  how  many  days  must  a  man  work  for 

Complex  Fractions. 

Art.  131.  To  reduce  complex  fractions  to  simple  ones; 
we  lirst,  Reduce  compound  fractions  to  simple  ones,  and  whole 
and  mixed  numbers  to  improper  fractions.  Then  consider 
the  denominator  of  the  complex  fraction  a  divisor  and 
proceed  as  in  division  of  fractions. 

3 

1.  Reduce  |  to  a  simple  fraction. 

5 

OPERATION. 

i  zrz  ?  X  X  =  "  =  If  Ans. 
2   Reduce  I  to  a  simple  fraction. 

3 
2. 

3.  Reduce  ^  to  a  simple  fraction. 

4 

-  of  ^ 

4.  Reduce  ^ ?  to  a  simple  fraction. 

5.  Reduce  5_2_I  to  a  simple  fraction. 

I  off 

6.  Reduce  — ?_  to  a  simple  fraction. 

ioff 

52- 

*l    Reduce 3—  to  a  simple  fraction. 

1  of  51 


A.RT.  132.]      LEAST  COMMON  MXLTIPLIE  OF  FRACTIONS.  113 

8   Reduce  ^^  ^^  ^^  to  a  simple  fraction. 
H  Of  3f 

J.  4-  1 
9.  Reduce  ^  ^  *  to  a  simple  fraction, 
a 

3  4 

10.  Reduce  ^  "^  ^  ^  ^  to  a  simple  fraction, 
f  ofl  +  f 

Least  Common  Multiple  of  Fractions. 

Art.  132,  The  Least  Common  Multiple  of  any  two 
or  more  fractions,  is  the  smallest  number  that  will,  when 
divided  by  each  of  them,  give  an  integer  for  a  quotient. 

Since,  in  dividing  any  number  by  a  fraction,  the  denomi- 
nator of  that  fraction  becomes  a  multiplier ^  and  the  nume- 
rator a  divisor  of  that  number ;  it  is  evident, —  Tha:t  the 
least  common  multiple  of  any  two  or  more  fractions,  after 
reducing  mixed  numbers  to  improper  fractions,  compound 
fractions  to  simple  ones,  and  all  to  their  lowest  terms^  will  be 
the  quotient  arising  from  dividing  the  least  common  multiple 
of  their  numerators  by  the  greatest  common  measure  of  their 
denomhmtors. 

1.  What  is  the  least  common  multiple  of  4f  f ,  SyV^,  and 

Solution. — The  above  mixed  numbers,  changed  to  improper 
fractions  become  ^-^^^  f^f,  and  2//.  These  fractions  when 
reduced  to  their  lowest  terms  become,  '^f,  Vi',  and  \y.  The 
least  common  multiple  of  the  numerators,  {65,  143,  and  117) 
of  these  fractions  is  6435.     See  Art.  94. 

The  greatest  common  measure  of  the  denominators,  (14, 
28,  and  42,)  of  these  fractions  is  14.  See  Art  89.  Hence, 
6435,  the  least  common  multiple  of  the  numerators  of  these 
fractions,  is  14  times  larger  than  the  least  common  multiple 
of  the  fractions.  Consequently  ^f|^  or  459y^j  is  the  least  com- 
mon multiple  required. 

2.  What  is  the  least  common  multiple  of  4y\,  1\  and  2/^? 

3.  What  is  the  least  common  multiple  of  4f ,  9yV>  and  12i? 

♦  Note.— The  student  will  readily  observe  that  the  Greatest  €ommom  Measukk 
of  any  two  or  more  fractions,  after  being  reduced  to  their  simplest  form,  will  be 
the  QUOTTEXT  arising  from  dividing  the  greatest  common  measure  oi  their  numerators 
by  the  least  common  multiple  of  their  denominators. 

What  is  the  greatest  common  measure  of  J,  ||,  and  IfP 


il4  FBiCTIONS.  [chap.    V, 

4.  What  is  the  least  common  multiple  of  }|,  |f ,  i|, 
and  f  f  ? 

5.  What  is  the  least  common  multiple  of  2|-,  16f ,  and 

10/3? 

PRACTICAL    QUESTIOXS    IN    MULTIPLES. 

1.  What  is  the  smallest  sum  of  money  for  which  a  per- 
son could  purchase,  either  a  number  of  geese,  at  $1^  a 
piece  ;  or  a  number  of  turkeys,  at  $2^  a  piece,  and  how- 
many  of  each  could  be  bought, — the  entire  sum  to  be 
employed  in  either  purchase  ? 

2.  A  can  travel  6|  miles  in  a  day;  B  lly^j  miles  ;  0 
20/0  miles  ;  and  D  SOa^j  miles  in  a  day.  What  is  the  least 
number  of  miles  that  will  afford  a  number  of  whole  days^ 
travel  for  any  of  the  four,  and  how  many  days  would  it 
take  each  to  accomplish  the  journey  ? 

3.  What  is  the  least  number  of  bushels  of  grain  that 
will  fill  a  number  of  hogsheads,  each  containing  10/j 
bushels;  a  number  of  boxes,  each  containing  23|f  bushels; 
or  a  number  of  bins,  each  containing  25 }|  bushels;  and 
how  many  times  would  it  fill  each  of  them  ? 

4.  What  is  the  smallest  sum  of  money  for  which  I  could 
purchase  a  number  of  cows,  at  $20^^;  a  number  of  oxen, 
at  $47|^;  or  a  number  of  horses,  at  $51y^j;  and  what 
number  of  each  could  I  purchase  for  that  sum  ? 

5.  Three  vessels  A,  B,  and  C  start  from  the  same  place, 
at  the  same  time,  and  sail  in  the  same  direction  around 
an  island  30  miles  in  circumference;  A  at  the  rate  of  3, 
B  11,  and  C  23  miles  an  hour.  How  many  hours  before 
they  will  all  meet  at  the  place  from  which  they  started  ? 
How  many  hours  before  they  will  first  meet,  and  at  what 
point  ?  Suppose  they  continue  sailing,  how.  often  will  they 
all  be  together  ? 

Solution. — A  moves  at  the  rate  of  3  miles  an  hour; 
consequently,  to  move  1  mile  it  will  take  ^  of  an  hour,  and 
to  move  30  miles,  (once  around  the  island,)  it  will  take 
y  or  10  hours.     In  a  similar  way,  we  find  B  will  move 


ART.  154.]       PRACTICAL  QUESTIONS  IN  MULTIPLES.  Il5 

once  around  the  island  in  ff  of  an  hour;  and  C,  in  ff  of 
an  hour.  Now  it  is  evident  that  the  least  common  multiple 
of  10,  f f,  and  |f  will  express  the  number  of  hours  that 
must  elapse  before  they  will  all  meet  at  the  place  from 
which  they  started  ;  which  is  30.  hours. 

How  long  before  they  will  first  meet  ?  C  gains  on  B, 
23  —  11=  12  miles  in  1  hour;  consequently,  to  gain  1  mile 
it  will  take  -^-^  of  an  hour,  and  to  gain  30  miles,  (the  dis- 
tance he  must  gain  before  he  overtakes  B,)  30  times  -^^  = 
^f ,  or  f  of  an  hour. — B  gains  on  All  —  3  =  8  miles  in 
1  hour;  hence  to  gain  1  mile  it  will  take  |  of  an  hour,  and 
to  gain  30  miles,  (the  distance  he  must  gain  before  he 
overtakes  A,)  30  times  |  =  y,  or  \^of  an  hour.  Since 
C  will  overtake  B  in  f  of  an  hour;  and  B  will  overtake 
A  in  y  of  an  hour;  it  is  evident  that  the  number  of  hours 
that  must  elapse  before  C  will  overtake  B,  at  the  same 
time  that  B  overtakes  A  will  be  the  least  common  multiple 
of  f  and  y ,  which  is  7^  hours. 

If  they  are  together  in  t^  hours, 
A  must  sail    22i  miles,  or  0  times  around  the  island  +  22^  miles. 
B    "       '*      82^     "      or  2     "  "  "      '*      +  22i     " 

C     "       "     172i     "•     or  5     "  ♦*  "      *'      +22i     « 

Consequently  22^  miles  from  the  place  from  which  they 
started  is  the  place  where  they  first  will  be  together. 

If  they  continue  traveling  they  will  be  together  every 
t|  hours. 

6.  If  four  men  A,  B,  C,  and  D,  start  from  the  same 
place  at  the  same  time,  and  walk  around  an  island  2t 
miles  in  circumference;  A  at  the  rate  of  4,  B  12,  C  20, 
and  D  28  miles  a  day;  how  many  miles  will  each  have  to 
travel  before  they  meet,  and  how  many  days  before  they 
will  all  meet  at  the  place  from  which  they  started  ? 

7.  There  are  three  wheels,  A,  B,  and  C,  each  lOf  feet 
in  circumference,  standing  with  their  axes  in  a  right  line, 
with  a  letter  M  on  the  circumference  of  each,  which  are 
also  in  a  right  line.  If  these  wheels  be  set  in  motion ;  A 
at  the  rate  of  5|,  B  7^,  and  C  131  feet  in  1  second,  how 
long  before  the  M's.  on  the  ch-cumference  of  the  wheels 


116  FRACTIONS.  [chap.  V. 

will  all  be  in  a  right  line  again,  and  how  many  revolutions 
will  each  have  made  ? 


PKACTICAL    QUESTIONS   IN   FRACTIONS. 

1.  Reduce  23 ly^  to  an  improper  fraction. 

2.  Reduce  478yVy  to  an  improper  fraction. 

3.  Reduce  -f  f  ^  to  a  mixed  number. 

4.  Reduce  2-iiS-  to  a  mixed  number. 

5.  Reduce  ^^  to  its  lowest  terms. 

6.  Reduce  |||ff  to  its  lowest  terms. 

*r.  Reduce  |-  of  f  of  yV  of  |f  of  ^f  to  its  simplest  form. 

8.  Reduce  |f  of  if  of  f  |  of  f  f  to  its  simplest  form. 

9.  Reduce  |,  f ,  |,  and  -f  to  equivalent  fractions,  hav- 
ing a  common  denomination. 

10.  Reduce  y\,  |f  and  \l  to  equivalent  fractions,  hav- 
ing the  least  common  denomination. 

11.  What  is  the  sum  of  |,  |,  |,  f,  and  ii  ? 

12.  What  is  the  sum  of  y^o  of  4^,  |  of  3|,  and  ^  of  6|  ? 

13.  From  8f  subtract  6f . 

14.  From  |  of  8^  subtract  f  of  2f . 

15.  Divide  3^  by  41-. 

16.  Divide  |  of  2|  by  |  of  4f . 

IT.  A  has  4f  times  $25,  and  B  has  ^  times  $8|;  how 
much  more  has  A  than  B. 

18.  A  has  I  of  $16;  Bfof$4T3;  Cf  of  $8621;  andD 
I  of  f  of  f  of  $168|.  How  many  dollars  have  they  together. 

19.  A  had  |  of  f  of  12^  times  $8643^,  and  paid  f  of 
■|-  of  it  for  a  farm ;  how  much  had  he  remaining  ? 

20.  A  had  $864T2,  which  was  6f  tpimes  as  much  as  B 
had;  how  much  had  B  ? 

21.  A  has  1278  sheep,  which  is  188  more  than  |  of  3^ 
times  B's  number;  how  many  sheep  has  B  ? 

22.  A  and  B  own  680  acres  of  land;  |  of  A^s  number 
of  acres  equals  f  of  B's; — how  many  acres  have  each  ? 

23.  A  farmer  has  495  bushels  of  wheat  and  rye  together. 


f 


ART.    132.]  PRACTICAL   QUESTIONS.  lit 

and  f  of  the  number  of  bush,  of  wheat  equals  ^  of  the  num- 
ber of  bush,  of  rye.     How  many  bushels  of  each  has  he  ? 

24.  A  speculator  bought  688  geese  and  turkeys;  how 
many  of  each  did  he  buy,  providing  there  were  only  |  as 
many  geese  as  turkeys  ? 

25.  A  and  B  together  own  824  sheep;  how  many  has 
each,  providing  A  has  1|  times  as  many  as  B  .? 

26.  A  owns  j\  of  a  certain  tract  of  land,  containing 
98600  acres;  B  owns  ^f  of  the  remainder;  C  owns  j\  as 
much  as  A  and  B  together;  and  D  owns  the  remainder. 
How  much  does  each  own  ? 

27.  A  merchant  expended  $463  for  dry  goods;  |  of 
^j  of  the  remainder  for  groceries;  and  what  then  re- 
mained, which  was  $4680,  he  expended  for  a  store  and 
lot.     How  much  did  the  groceries  cost  ? 

28.  A  gentleman  invested  |  of  his  fortune  in  specula- 
tion, and  the  remainder,  which  was  $1630  more  than  the 
half  of  his  fortune,  he  put  out  on  interest.  At  the  end 
of  the  year  he  gained  by  speculation  y\  as  much  as  he 
laid  out,  and  his  interest  was  /^  of  the  principal;  how 
much  was  his  fortune,  and  how  much  did  he  gain  during 
the  year  ? 

29.  A  man  being  asked  the  value  of  his  horse,  replied, 
that  its  value  increased  by  its  -f  and  $268|  more,  equaled 
$864.     What  was  the  value  of  the  horse  ? 

30.  A  farmer  has  -f  of  the  number  of  his  sheep  in  one 
field  ;  and  the  remainder,  which  is  46  more  than  the  half 
of  his  flock  in  a  second  field.  How  many  sheep  has  he  in 
each  field,  and  how  many  in  both  ? 

31.  A  certain  sum  of  money  was  divided  between  two 
brothers,  James  and  Jackson  ;  James  took  |  of  it,  lacking 
$145  ;  and  Jackson  the  remainder.  It  now  appears  that 
each  has  the  same  sum.     How  much  did  each  receive  ? 

32.  From  a  certain  flock  of  sheep  A  purchased  f  of 
them  ;  B  f  of  them  ;  C  f  of  them  ;  and  D  the  remain- 
der, which  was  116.  How  many  sheep  were  then  in  the 
field,  and  how  many  did  A,  B,  and  C,  buy  respectively  ? 

33.  A  has  |  of  1-f  times  $2660  which  is  3^  times  as 
many  again  as  B  has.     How  many  dollars  has  B  ? 


,118  FRACTIONS.  [chap.  V, 

34.  A  and  B  together  own  a  farm  ?  A  owns  y\-  of  it  ; 
and  B  /_  of  it.  If  B  should  sell  to  A  12|  acres,  they 
would  then  each  have  the  same  number  of  acres.  How 
many  acres  has  each  ? 

35.  An  estate  was  divided  among  A,  B,  and  C.  A 
had  ^  of  it ;  B  ^  of  it  ;  and  C  the  remainder.  A,  by 
this  division,  received  $180  more  than  B.  How  much 
was  the  estate,  and  how  much  did  each  receive  ? 

36.  Divide  $9926  among  A,  B,  C,  and  D,  so  that  A 
shall  have  -f  of  it,  lacking  $1812  ;  B  |  of  the  remainder, 
lacking  $858  ;  C  f  of  what  now  remains,  lacking  $1880  ; 
and  D  what  is  left  ? 

3t.  A  gentleman's  house  cost  $4800,  and  If  times  its 
cost,  is  3i  times  -f  of  the  cost  of  the  furniture  contained 
in  it  ;  what  was  the  cost  of  the  furniture  ? 

38.  Henry  had  f  of  a  certain  fortune  ;  Perry  ^^  Of  it ; 
and  Elisha  the  remainder,  which  was  $1600.  How  much 
was  the  fortune,  and  how  much  did  Henry  and  Perry  re- 
ceive respectively  ? 

39.  Bought  at  one  time  460  acres  of  land,  at  $25|-  an 
acre  ;  at  another  time  345  acres,  at  $43^  an  acre.  If  | 
of  the  whole  quantity  were  sold,  at  $21  an  acre,  and  the 
remainder,  at  $34  an  acre,  what  would  be  the  gain  or  loss  ? 

40.  A  merchant  purchased  120  yards  of  cloth  for  $780, 
and  sold  |  of  it  at  a  profit  of  $lf  a  yard ;  and  the  re- 
mainder, at  a  loss  of  $f  a  yard.  How  much  did  he  gain 
by  the  operation  ? 

41.  A  person  bought  38  barrels  of  flour  at  $4|  a  bar- 
rel. Having  sold  21|-  barrels  of  them,  at  $4f  a  barrel,  at 
what  price  a  barrel  must  the  remainder  be  sold  to  gain 
$25^  on  the  whole. 

42.  James,  Henry,  and  Joseph  were  employed  to  hoe  a 
field  of  corn  for  $32.10.  James  could  hoe  a  row  in  2 Of 
minutes;  Henry  in  25|  minutes;  and  Joseph  in  32y^o  min- 
ntes.  It  so  happened  that  when  they  all  first  came  to  the 
end  of  a  row  at  the  same  instant,  that  the  work  was  com- 
pleted. How  long  were  they  engaged  in  the  field;  how 
many  rows  did  the  field  contain;  and  how  m'ich  in  equity 
ought  each  to  receive  ? 


ART.    134.]  DECIMAL   FRACTIONS.  119 

CHAPTER  YI. 
DECIMAL   FRACTIONS. 


Art.  133.  A  Decimal  Fraction  is  one  in  which  the 
denominator  is  not  expressed,  but  is  understood  to  be  a 
unit  followed  by  one  or  more  ciphers  ;  or  such  a  fraction 
the  successive  orders  of  which  increase  from  right  to  left  in 
a  tenfold  ratio,  consequently  decrease  from  left  to  right  in 
the  same  ratio. 

Decimal  Fractions  originate  from  dividing  1  into  10 
equal  parts  ;  each  of  these  parts  into  10  other  equal  parts  ; 
and  each  of  the  parts  thus  obtained  into  10  other  equal 
parts,  and  so  on,  indefinitely.  Thus,  \  —  10  =  yV  ;  yV  "^ 
10  =  y^o  ;  y^o-  -h  10  =  yoVoj  &c.,  wMch  are  expressed 
in  Decimals  as  follows  : — 

tV  =  -1  ;  T^o  =  01  ;  y^Vo  =  -001,  &c. 

Art.  134.  In  expressing  Decimal  Fractions,  the  nu- 
merator only  is  written  with  a  point  before  it,  called  a 
Decimal  'point  or  Separatrix,  to  distinguish  it  from  whole 
numbers*;  the  denominator  being  understood.     Thus, 

y\  =  't         tenths. 
Too  =  '^^       hundredths. 
To\o  =  "00 1     thousandth's. 


_7 

0  0  0  0 


=  'OOOt  ten-thousandths. 
r=   0223  ten-thousandths. 


By  inspecting  the  above  fractions,  it  is  observed  that 
tenths  occupy  the  first  place  at  the  right  of  the  decimal 
point ;  that  hundredths  occupy  the  second  place;  that  thovr- 
sandths  occupy  the  third  place,  &c. 

We  also  observe  that  each  removal  of  a  figure  one  place 
towards  the  right,  decreases  its  value  in  a  tenfold  ratio. 
Hence, 


120  DECIMAL   FRACTIONS.  [cHAP.  VI. 

Every  cipher  placed  on  the  left  of  a  decimal  figure  dimin- 
ishes its  value  in  a  tenfold  ratio.  Thus,  '9  =  -^q,  '09  ^- 
rW,  aid  -009  =  y/oo,  &c. 

If  a  cipher  be  placed  on  the  right  of  a  decimal  figure, 
it  does  not  change  its  value,  as  the  figure  still  occupies 
the  same  place.     Thus,  '9  =  '90  =  "900,  y\  =  rVo  = 

JULQ_    Jirn 
To  0  0>  *^^' 

^  Numeration  of  Decimal  Fractions. 

Art.  135.  A  whole  number  and  a  decimal  fraction, 
when  considered  together,  is  called  a  mixed  number ;  the 
relation  and  names  of  which  can  be  learned  from  the  fol- 
lowing 

TABLE. 


<r 

OQ 

'S 

a 

A 

.      m        . 

s 

■4^ 

i 

-5 

i 

1 

rd 

1 

1 

1 
d 

1 

isand 
dreds, 

3. 

i  1 

^ 

3 

S 

rd        r3 

.1  s 

■-d 

m 

,d 

d 
o 

.2 

-3 

■73 

c§ 

J   ^    §    § 

•3  1 

1 

O 

d 

d 
d 

d 
s 

3 

d 

d 

d 

i 

H    ffi    H 

^  1 
4 

H 

ffl 

H 

H 

ffi 

ffi 

PQ 

H 

K 

9    3     1 

3 

3 

3 

3 

3 

4     7 

8 

6 

2 

7 

Whole  Numbers. 

Decimals. 

Art.  136.  To  read  a  Decimal  number  expressed  in 
figures. 

Read  the  figures  as  in  whole  numhcrs,  and  add  the  name 
of  the  decimal  place.  Thus,  "9  is  read  nine-tenths ;  '09  is 
read  nine  hundredths  ;  '100024  is  read  one  hundred  thousand 
and  24  millionths,  &c 

Rkmark.— To  ascertain  the  name  of  the  right  hand  figure,  begin  at  the  left 
and  name  each  figure  till  you  come  t©  the  last  one,  which  will  be  the  name 
required. 


ART.    13t.]  UNITED    STATES    CURRENCY. 


121 


Read  the  following  numbers. 


1.  13 

2.  -06 

3.  -0102 

4.  -02202 


7. 


9. 


46-824 

36-4231 

61-46212 

141-8630202 


11.  .  102-460203 

12.  14683-04000602 

13.  18602-84683002 

14.  10001-861320401 


5.  -060108    10.     1468-10002   15.    1010101-10101010101 

United  States  Currency,*  or  Federal  Money. 

Art.  137.  United  States  Currency  is  the  legal  Money 
of  the  United  States,  and  is  expressed  according  to  the 
Decimal  Scale  of  notation. 

Money  may  be  considered  the  measure  of  the  value  of 
things. 

f  The  denominations  of  the  United  States  Currency  are 
Eaf^hs,  Dollars,  Dimes,  Cents,  and  Mills. 

The  coins  of  the  United  States  are  the 
Double-Eagle, 


made  of  gold, 


►  made  of  silver; 


Eagle 

Half-Eagle, 

Quarter-Eagle, 

and  Dollar, 

The  Dollar, 

Half-Dollar, 

Quarter-Dollar, 

Dime, 

Half-Dime, 

and  Three-cent  piece. 

The  cent  is  made  of  copper. 

The  mill  is  not  coined. 

Note. — By  an  act  of  Congress,  January  ISth,  1837,  the  gold 
and  silver  coin  must  consist  of  y^^  pure  metal,  and  -^-^  alloy. 

Remark. — *  This  currency  is  usually  called  Federal  Muney,  because  it  was 
made  the  currency  of  the  United  States  at  the  time  of  the  adoption  of  the 
Federal  Constitution,  August  8th,  1786. 

t  The  word  Dollar  is  derived  from  a  Danish  word,  which  was  derived  from 
Dale,  the  name  of  the  town  in  which  this  coin  was  first  made.  The  word 
Dime  is  derived  from  a  French  word  signifying  ten; — the  word  Cent,  from  a 
Latin  word  signifying  one  hundred; — the  word  Mill,  from  a  Latin  word  signi- 
fying one  thousand.  The  terms,  Dime,  Cent,  and  Mill,  are  applied  to  coins  of 
our  currency,  in  consequence  of  the  relation  they  respectively  bear  to  the 
Dollar. 

6 


122  DECIMAL   FRACTIONS.  ^CEAT.  VI. 

The  alloy  for  gold  must  consist  of  an  equiil  quantity 
of  silver  and  copper, .  and  the  alloy  for  silver  of  pure 
copper. 

The  three-cent  piece  is  |  silver  and  |  copper. 

Before  183t,  the  gold  for  coinage  consisted  of  ||  pure 
I  gold,  -^j  silver,  and  2V  copper;  or,  as  sometimes  expressed, 
22  carats  gold,  1  of  silver,  and  1  of  copper;  the  word 
ca7-at  meaning  one  twenty-fourth. 

Silver  for  coinage  consisted  of  1489  parts  of  pure  silver, 
and  179  parts  of  pure  copper;  or,  expressed  in  carats, 
2IT3V  of  silver,  and  2//^  of  copper. 

Table  of  United  States  Currency. 


10  Mills 

make 

1  Cent, 

marked 

C. 

10  Cents 

C( 

1  Dime, 

u 

d. 

10  Dimes 

(I 

1  Dollar, 

<( 

$.* 

10  Dollars 

« 

1  Eagle, 

u 

E. 

Art.  138.  The  accounts  of  the  United  States  are 
kept  in  Dollars  Cents,  and  Mills.  In  business  transactions, 
the  eagle  is  expressed  in  dollars,^-the  dime  in  cents.  Thus, 
4  eagles,  according  to  the  preceding  table,  is  =  40  dol- 
lars; and,  instead  of  saying  4  eagles  and  5  dollars,  we 
say  45  dollars.  Also,  *I  dirn^s  are  =  70  cents;  therefore, 
instead  of  saying  7  dimes  and  9  cents,  we  say  79  cents,  &c. 

Art.  139.  In  the  United  States  Currency,  dollars  are 
Integers,  and  therefore,  occupy  the  place  of  units.  Cents 
express  hundredths  of  a  dollar,  consequently  they  occupy 
the  first  and  second  place  on  the  right  of  the  decimal 
point;  the  third  place  is  mills;  &c.  Thus,  $47'356,  is 
read  forty-seven  dollars  thirty-five  cents  and  six  mills. 

Art.  140.  To  express  any  number  of  cents  less  than 
10,  there  should  be  a  cipher  placed  between  them  and  the 
decimal  point,  as  cents  are  hundredths  of  a  dollar,  and 
therefore  occupy  two  places;  if  mills  only  are  expressed, 


*  This  symbol  is  probably  a  contraction  of  the  letter  U,  placed  upon  an 
8,  to  denote  U.  S.  (United  Stateo.) 


ABT.  142.]  REDUCTION  OF  DECIMALS  TO  COMMON  FRACTIONS.  123 

two  ciphers  should  be   placed   between    them   and   the 

decimal  point.     Tiuis: 

4  cents  is  written  $*04,  and  is  read  4  hundredths  of  a  dollar. 
12^  cents  is  written  $12^,  and  is  read  12^  hundredths  of  a  dollar. 
I  ota  cent  is  written  $-00f ,  and  is  read  |  of  1  hundredths  of  a  dol- 
lar. 5  mills  is  written  |-005,  and  is  read  5  thousandths  of  a 
dollar.  4  cents  and  6  mills  is  written  $-046,  and  is  read  46 
thousandths  of  a  dollar. 

Art.  141.  It  might  be  well  to  observe,  that  the  first 
figure  after  the  decimal  point,  expresses  te^iths  of  a  dollar, 
or  tens  of  cents  ;  the  second,  cents;  the  first  two  places 
taken  together  express  hundredths  of  a  dollar,  or  cents ; 
the  third  viills  ;  the  fourth  tenths  of  mills,  &c. 

Reduction  of  Decimals  to  Common  Fractions. 

Art.  142,  From  what  has  been  said  of  Decimal  FraC' 
tions,  we  infer  that  a  decimal  can  be  reduced  to  a  common 
fraction  by  erasing  the  decimal  pointy  and  underneath  writing 
the  denominator,  which  is  a  unit  followed  by  as  many  ciphers 
as  there  are  places  in  the  decimal ;  then  reduce  the  fraction 
to  its  lowest  terms. 

I.  Reduce  '025  to  a  common  fraction. 

operation. 

•025  =  yV/o  =  tV-  Ans. 
Reduce  each  of  the  following  decimals  of  a  dollar  to 
equivalent  common  fractions. 

2.  $0625.  5.  $-25.  8.  $-625. 

3.  $-125.  6.  $-5.  9.  $'875. 

4.  $-375.  1.  $-75.  10.  $-5625. 

II.  Express  6*0625  by  an  integer  and  a  common  frac- 
tion. 

12.  Express  12  25  by  an  integer  and  a  common  fraction 

13.  Express  145  by  an  integer  and  a  common  fraction 

14.  Express  25'625  by  an  integer  and  a  common  frac 
tion. 

15.  Express  45'8t6  by  an  integer  and  a  common  fraoi 
tion. 


124 


DECIMAL   FRACTIONS. 


[chap.  VI. 


16.  Express  3t*15  by  an  integer  and  a  common  frac- 
tion. 

It.  Express  16 'St 5  by  an  integer  and  a  common  frac« 
tion. 

18.  Express  34"93t5  by  an  integer  and  a  common 
fraction. 

19.  Express  62-5636  by  an  integer  and  a  common 
fraction. 

20.  Express  141*18t5  by  an  integer  and  a  common 
fraction. 


Reduction  of  Common  Fractions  to  Decimals. 
1.  Reduce  -f  to  a  decimal. 


OPERATION 

• 

. 

.d 

X 

■S 

'2 

g    c3 

n 

1 

c 
5 

S  .a 

j3 

o 

5 

s 

Xi 

m 

1 

a 

o 

J3 

^    6 

i2 

•5 

3 

H 

Ti  <a 

C 

H 

C 

a 

a 

'S 

(U 

3 

j3 

s 

t3 

H 

K 

H 

H 

W 

7)4. 

•0 

0 

0 

0 

0 

0 

5 

7 

1 

4 

2+* 

Explanation. — 7  is  not  contained  into 
4  units  a  whole  number  of  times,  there- 
fore, we  reduce  4  to  tenths;  4  =  40 
tenths.  7  is  contained  in  40  tenths,  5 
tenths  times  and  5  tenths  remaining, — 
which  equals  50  hundredths.  7  is  con- 
tained in  50  hundredth,  7  hundredtJis 
times,  and  1  hundredth  remaining,  which 
equals  10  thousantlis^  &c. 


Remark. — If  the  decimal  is  carried  out  six  places,  the  result  will  be  suffl. 
ciently  exact  for  all  practical  purposes. 

What  decimal  of  a  dollar  is  equal  to  the  following 
fractions  of  a  dollar  ? 


4.  $f. 


5.  $f. 


6.  $f. 


1.  %h 

2.  $1 

3.  $^. 

8.  Reduce  f  12^  to  an  equivalent  decimal  expression 

9.  Reduce  $16^  to  an  equivalent  decimal  expression. 


*  The  symbol,  -f-;  when  plficed  after  a  decimal  indicates  more. 


ART.    144.]  REPETENDS.  125 

10.  Reduce  $42|  to  an  equivalent  decimal  expression. 

11.  Reduce  $25 y\  to  an  equivalent  decimal  expression 

12.  Reduce  llS^'^g-  to  an  equivalent  decimal  expression, 

13.  Reduce  |3t||  to  an  equivalent  decimal  expression. 

Reduction  of  Mixed  Decimals  to  Simple  Decimals. 

Art.  143.  If,  in  the  place  of  the  Common  Fraction, 
we  place  its  equivalent  decimal  without  the  point  prefixed, 
the  value  remains  the  same.     Thus,  .4f . 

The  f  =  -75,  hence  '41  =  -475 

14.  Reduce  '251  to  an  equivalent  simple  decimal. 

15.  Reduce  •2t-]-  to  an  equivalent  simple  decimal. 

16.  Reduce  "SSi  to  an  equivalent  simple  decimal. 
It.  Reduce  '411  to  an  equivalent  simple  decimal. 

18.  Reduce  •64f  to  an  equivalent  simple  decimal. 

19.  Reduce  '841  to  an  equivalent  simple  decimal. 

20.  Reduce  "1461  to  an  equivalent  simple  decimal. 


Repetends. 

Art.  144.  If  a  common  fraction  cannot  be  accurately 
expressed  in  decimals,  it  is  evident  that  the  decimal  figures 
will  recur  in  periods;  and  that  the  number  of  figures  in 
the  period  cannot  exceed  the  number  of  units  in  th6  denom- 
inator, less  one — for  every  remainder  must  be  less  than  the 
denominator;  and  whenever  a  remainder  occurs  like  one 
previously  obtained,  the  decimal  figures  will  begin  to 
repeat. 

Decimals  that  recur  in  this  way  are  called  repeating 
decimals ;  the  figures  repeated  are  called  a  repeteiid,  and  are 
distinguished  by  a  (')  placed  over  the  first  and  last,  as, 
1  =z  -333,  &c.  =  -3;  1  =  -142857142,  &c.  =  •142857. 

If  decimal  figures  precede  the  repetend,  they  are  called 
i\Q  finite  part  of  the  decimal;  as,  -Jj  =  0-0833,  &c.  The 
finite  part  is  -08. 


126  DECIMAL   FRACTIONS.  [CHAP.    VI. 

When  tbe  repeating  period  begins  with  the  first  decimal 
figure,  it  is  called  a  simple  repetend. 

A  simple  repetend  that  contains  as  many  figures  in  the 
repeating  part  as  there  are  units  in  the  denominator,  less 
one,  is  called  a  Perfect  Repetend.     Thus, 

\   =  0-142851:. 

tV  =  0-658823529411l64i. 

tV  =  0-6526315^894l36842i. 

&c.  &c. 

Compound  Repetends. 

The  following  fractions  are  called  Compound  Repetends^ 
because  they  consist  of  a  finite  and  a  repeating  part. 

i   =016. 

tV  =  0083. 

tV  =  0-0h4285. 

&c.     &c. 

Art.  145.  Repetends    Reduced  to  Common  Frations, 

•i  =  X;  -01  =  ir.  -ooi  =  ^1^;  -oooi  =  ^^v». 

•2  =  I;  -02  ^  -/^;  -002  =  «f^;  '6002  =  ^/^^. 
.     -3  =.  f ;  -03  =  ^V;  -003  =  ^|^;  -0003  =  ^^\^. 
&c.,     &c.,     &c.,     &c. 

From  which  we  observe  that  Uie  repetend,  with  the 
decimal  point  and  useless  ciphers  on  the  left,  erased,  is  the 
numerator,  and  the  denominator  is  as  many  9's  as  there  are 
places  in  the  repetend. 

Thus,  -004004,  &c.  =  ^f^. 

1.  Redcce  '142857  to  a  common  fraction. 

2.  Reduce  02*1  to  a  common  fraction. 

3.  Reduce  'Oli  to  a  common  fraction. 


JLPT.   147.  I       REDUCTION  OF  COMPOUND  REPETENDS  121 

4.  Reduce  •'12  to  a  common  fraction. 

5.  Reduce  •123  to  a  common  fraction. 

6.  Reduce  -238095  to  a  common  fraction. 

Art.  14G,     Reduction  of  Compound  Repetends 

7.  Reduce  'OSSSSS,  &c.,  to  a  common  fraction. 

OPERATION. 

•083333,  &c.  =  -083  =  -081  =  ^  =  ^  =  1.  Ana. 

100       300       12 

8.  Reduce  '16  to  a  common  fraction. 

9.  Reduce  '06  to  a  common  fraction. 

10.  Reduce  '045  to  a  common  fraction. 

11.  Reduce  '0416  to  a  common  fraction. 

12.  Reduce  •0'il4285  to  a  common  fraction. 

Art.  147.  We  have  se^  that  the  value  of  some  common 
fractions  can  be  accurately  expressed  in  decimals,  while 
that  of  others  can  only  be  approximately  expressed,  as  the 
process  of  division  will  never  terminate. 

As  we  annex  ciphers  to  the  numerator  and  divide  by 
the  denominator  to  change  a  common  to  a  decimal  fraction; 
and,  as  annexing  a  cipher  to  any  number  is  the  same  as 
multiplying  it  by  10,  it  follows  that  whenever  the  prime 
factors  of  the  denominator  of  a  common  fraction  (affer 
reducing  it  to  its  lowest  terms)  do  not  differ  from  2  and  5 
(the  prime  factors  of  10),  the  division  will  terminate. 

Hence,  to  determine  whether  a  common  fraction  can  be 
accurately  expressed  in  decimals, 

Reduce  it  to  its  lowest  terms,  then  resolve  the  denominator 
into  its  prime  factors  ;  if  these  factors  do  not  differ  from  2 
and  5  {the  factors  of  lOJ,  the  fraction  can  he  accurately 
expressed  in  decimals.  It  is  also  evident  that  the  highest  ex- 
yo'iie'nt  of  the  2  or  5  will  denote  the  number  of  decimal  places 
required  tt  express  it. 


128  DECIMAL  FRACTIONS.  [cHAP.   V 

1.  Cun  -gf  J  be  accurately  expressed  in  decimals  ?  If  so, 
how  many  places  will  be  required  to  express  it  ? 

2.  Can  yf  ^  be  accurately  expressed  in  decimals  ?  If  so, 
how  many  places  will  be  required  to  express  it  ? 

3.  Can  y|j  be  accurately  expressed  in  decimals  ?  If  so, 
how  many  places  will  be  required  to  express  it  ? 

4.  Can  -^  be  accurately  expressed  in  decimals  ?  If  so, 
how  many  places  will  be  required  to  express  it  ? 

6.  Can  -y Jo"  be  accurately  expressed  in  decimals  ? 

Addition  of  Decimals  and  United  States  Currency. 

Art.  148.  Since  decimals  increase  from  right  to  left, 
and  decrease  from  left  to  right,  in  the  same  ratio  as  simple 
numbers,  we  can  add,  subtract,  multiply,  or  divide  them,  in 
the  same  manner  as  though  they  were  abstract  numbers. 
In  addition,  observe  to  place  units  under  units,  tens  under 
tens,  &c.  Hence  the  decimal  points  will  always  come  one 
under  another. 

1.  What  is  the  sum  of  14-623,  231-6231,  101-36, 
8002-68023,  and  1-462312  ?      . 

operation. 


is    to 

'«  2 

.  <»■  S   2 

c  -a  TJ    c    o  TS  5 

Si's    wiS-'S    !=''^'S:§ 

14-6  2  3 
2  3  1-6231 
10  1-3  6 
8  0  0  2-68023 

7-462312 


835  7-7  48642  Ans. 

2.  What  is  the  sum  of  12-6,  63-04,  342-021,  8462-':321 
62-48132,  and  412-16321? 

3.  What  is  the  sum  of  $27046,  $14-4041,  $7-86321, 
$324-8631,  and  $412382631  ? 


ART.    148.J  PRACTICAL   QUESTIONS.  129 

4.  What  is  the  sum  of  8-321,  9-6231,  84-n63,116-84324, 
and  18312.83201  ? 

5.  What  is  the  sum  of  82-631,  l-t632,  8413-0001, 
t32-46n,  842-n,  and  9802  ? 

PRACTICAL    QUESTIONS. 

1.  Sold  a  box  of  candles  for  $12*25  ;  a  barrel  of  flour 
for  $t-t5  ;  a  sack  of  coffee  for  $12121  j  and  a  barrel  of 
sugar  for  $18-3t|  ;  required  the  amount  I  should  receive, 

2.  What  cost  a  horse,  a  yoke  of  oxen,  a  cow,  and  a 
sheep,  if  the  horse  cost  $141'62i;  the  oxen,  $184-06^; 
the  cow,  $46-82;  and  the  sheep,  $7-06i? 

3.  A  merchant  bought  broadcloth  to  the  amount  of 
$8t2-45;  muslin  and  linen  to  the  amount  of  $184*75; 
sugar  to  the  amount  of  $296*85  ;  and  flour  .to  the 
amount  of  $38t-80.     What  was  the  whole  cost  ? 

4.  A  gentleman  has  some  young  cattle  worth  $6t82-8t; 
a  horse  worth  $285-60  ;  a  yoke  of  oxen  worth  $196-87; 
four  cows  worth  $210-20;  and  a  farm  worth  $8642-84. 
What  is  the  value  of  the  whole  ? 

5.  A  merchant  bought  a  quantity  of  goods  for 
$89785;95  ;  paid  for  duties  $897*40;  and  for  transporta- 
tion $38887.  For  how  much  must  he  sell  the  goods  to 
gain  $346-82  ? 

6.  Bought  a  ton  of  hay,  for  $14-87|  ;  a  cord  of  wood, 
for  $6-12i;  a  barrel  of  apples,  for  $3-06^;  a  barrel  of 
flour,  for  $8-371;  and  a  quarter  of  beef,  for  $9*87i.  Re- 
quired the  sum  to  be  paid  ? 

7.  A  farmer's  bill  at  the  store  was  as  follows  :  4i  yards 
of  cloth,  $24-121;  3  pair  of  boots,  $16-87^;  a  dozen 
skeins  of  silk,  $*87i;  and  15  yards  of  muslin,  $l*12i. 
Required  the  amount  of  the  bill  ? 

8.  A  farmer  sold  produce  as  follows:  wheat,  for 
$325-871  ;  corn,  for  $137*621;  rye,  for  $237*85;  oats, 
for  $96061;  hay,  for  $62-62i.  Required  the  amount  of 
the  sale  ? 

9.  What  should  be  paid  for  a  barrel  of  sugar,  worth 
$18-471;. a  quarter  of  beef,  worth  $9*871;  a  barrel  of 
flour,  worth  $6*37^;  a  box  of  raisins,  worth  $8*20;  a  fir 


13U  I>ECIMAL    FRACTIONS.  [cHAP.    VI. 

kin  of  butter,  worth  $15'9t|;  and  a  barrel  of  molasses, 
worth  $12-25? 

10.  Bought  a  quantity  of  sugar,  for  $183'92  ;  a  quan- 
tity of  flour,  for  $227-621;  a  quantity  of  hams,  for 
$384'18f;  a  quantity  of  co*rn,  for  $38()-8Ti.  For  how 
much  must  it  all  be  sold  so  as  to  gain  $465-85,  after  pay- 
ing $120-37^  for  cartage  and  storage  ? 


Subtraction  of  Decimals  and  the  United  States 
Currency. 

1.  From  64-5  subtract  37-8046. 

OPERATION. 

Min.  64-5000 
Sub.  37-8046 

Rem.  26-6954 

Rkmark.— In  examples  of  this  kind  we  annex  ciphers  to  the  minuend,  which 
does  not  eftect  its  value.     (See  last  para^^raph,  Art.  134.) 

Care  must  be  taken  to  place  the  numbers  so  that  the  decimal  points  shall 
stand  one  under  another,  in  order  that  units  may  be  taken  from  units,  &c.j 
tenths  from  tenths,  &c. 

2.  From  204*614  subtract  9-131. 

3.  From  6  subtract  4-00006. 

4.  From  4-4  subtract  3-00004. 

5.  From  1  subtract  -000001. 

6.  From  16802-4682  subtract  981-8364. 

7^  Subtract  10014-40001  from  80084-600861. 

practical  questions. 

1,  A  man  bought  a  span  of  horses  for  $465"85,  and  a 
yoke  of  oxen  for  $195-38  ;  how  much  more  did  he  pay 
for  the  horses  than  for  the  oxen  ? 

2.  A  gentleman  having  $18654-84,  gave  $2685-69  of  it 
for  a  store  ;  how  much  money  has  he  remaining  ? 

3  A  man  is  owing  $6785-95,  and  has  due  him  $9986-125; 
how  much  more  is  due  him  than  what  he  owes  ? 

4.  A  quantity  of  lumber  was  bought  for  $5682-18|,  and 
sold  for  $7631-561;  how  much  was  the  gain .- 


ART.     149. J  MULTIPLICATION    OF    DECIMALS.  131 

6.  A  quantty  of  flour  was  purchased  for  $3896-12^, 
and  sold  for  $oJ:'9-18f  ;  how  much  was  the  loss  ? 

6.  A  grazier  uonght  cattle  for  $384'95,  and  sheep 
for  $135-68.  He  sold  the  cattle  for  $419-12^,  and  the 
sheep  for  $109*72^;  how  much  did  he  gain  by  these  trans- 
actions ? 

7.  A  manufacturer  purchased  a  quantity  of  cotton  for 
$387  95,  which  he  made  into  cloth,  at  an  expense  of 
$184*06|  ;  how  much  will  he  make  by  selling  the  cloth 
for  $600  ? 

8.  A  speculator  purchased  wheat  for  $587 -871,  and 
pork  for  $968' 12|.  He  sold  his  wheat  for  $73918^,  and 
his  pork  for  $78437^.  Did  he  gain  or  lose  by  the  opera- 
tion, and  how  much  ? 

9.  A  speculator  bought  at  one  time  347 '35  acres  of 
land;  at  another,  637*25  acres;  and  at  another,  1435'7 
acres.  He  is  desirous  of  making  his  purchases  amount  to 
1 225*5  acre's.     How  much  land  does  he  still  want  ? 


Multiplication   of   Decimals   and   the   United    States 
Currency. 

Art.  149.  One-tenth  taken. two  times,  or  multiplied 
by  2,  gives  for  a  product  y^^;  if  taken  once,  or  multiplied 
by  1,  the  product  will  be  -,V;  if  taken  one-tenth  of  a  time, 
or  multiplied  by  j\  of  1,  the  product  must  be  y^  of  yV 

—  _X_. thn«       1    V  Jl—  —  _J •    _i_  V  — '_  — 1 • 1 V  _' 

10  0)  HJUO,     10-^10  100)      100^^10  1000)1000'^lir 

=  Toooo»  &c->  which  decimally  expressed  becomes  'IX'l 
=  01;  -OlX'l^-OOl;  "001 X  •bl  =  -00001,&c.  From  which 
we  observe  that  the  number  of  decimal  places  in  the  pro- 
duct is  equal  to  the  number  of  ciphers  (which  in  practice 
is  understood)  in  the  denominators  of  both  factors,  which 
/s  always  equal  to  the  number  of  decimal  places  in  the 
two  factors.  Hence  to  multiply  one  decimal  by  another 
we  proceed  as  in  whole  numbers,  and/rom  the  right  of  t/ie 
'product,  point  off  as  many  places  for  dccim.als  as  there  are 
decimal  places  in  both  multiplier  and  multiplicand.  Should 
there  not  be  places  enough  in  the  product,  prefix  ciphers. 


132 


DECIMAL   FRACTIONS. 


[chap.  VI. 


1.  Multiply  4-86  by  t-39. 

2.  Multiply  14-683  by  10-83. 

3.  Multiply  122-  by  46-7832. 

4.  What  is  the  product  of  202*002  and  1  0002  ? 

5.  What  is  the  product  of  165-3701  and  47-8201  ? 

6.  What  is  the  product  of  3786-478  and  831-0241  ? 

7.  What  is  the  product  of  8602-8312  and  48-76324  ? 

Art.  150.     A  decimal  is  multijplied  hy  10,   100,  1000, 
SfC.y  by  merely  reinoving  the  decimal  point  as  many  places  to 
the  right  as  there  are  ciphers  in  the  multiplier.     If  necessary 
auTiex  ciphers  to  the  number. 
C      86-723    ) 
Multiply   I      14-243    }  by  10. 
r    1001-001  S 


Multiply 


Multiply 


Multiply 


:    8-076 

41-3421 
'  716-311' 

,    1-832    ^ 

30-12    ' 

'  8-63412  ' 

148-63 
7-34876 
28-31017 
186-4 

2-7 


by  100. 


by  1000. 


1^  by  10000. 


PRACTICAL    QUESTIONS. 

1.  What  cost  95  tons  of  hay,  at  $12-75  a  ton  ? 

2.  What  cost  125  yards  of  broadcloth,  at  $5 -37  J  a 
yard  ? 

3.  What   cost   275   bushels   of  potatoes,   at   $'62^  a 
bushel  ? 

4.  What  cost  384  barrels  of  sugar,  at-  |17-87^  a  bar- 
rel ? 

5.  What  cost  312  pounds  of  butter,  at  $18^  a  pound  ? 

6.  What  cost  245  barrels  of  molasses,  at  $23-18|  a 
barrel  ? 


ART.    151.]  DIVISION,  OF   DECIMALS.  138 

1.  If  25  men  earn  $3*T'18f  in  one  day,  how  much 
can  they  earn  in  a  year,  of  365  days  ?  (not  counting  Sun- 
days. 

8.  A  gentleman  purchased  a  farm  containing  445-5 
acres,  at  $34 "12^  an  acre  ;  how  much  did  the  farm  cost 
him  ? 

9.  How  much  should  be  paid  for  25*5  cwt.  of  tobacco, 
at  $12-37i  a  hundred  weight  ? 

10.  Bought  275  sheep,  at  $1-87^  a  head,  and  sold  them, 
at  $2-12^  a  head;  how  much  did  I  gain  by  the  opera- 
tion ? 

Division  of  Decimals  and  the  TJnited  States  Currency. 

Art.  151.  The  quotient  arising  from  dividing  any  num' 
her  by  another  of  the  same  denominxition,  is  a  whole  number. 
Thus,  if  units  be  divided  by  units,  tenths  by  tenths,  hun- 
dredths by  hundredths,  or  thousandths  by  thousandths, 
&c.,  the  quotient  will  be  a  whole  number.  Therefore,  in 
the  division  of  decimals,  when  the  divisor  and  dividend 
each  contain  the  same  number  of  decimal  places,  the  quo- 
tient will  be  a  whole  number;  and  if  the  dividend  contain 
more  decimal  places  than  the  divisor,  there  must  of  neces- 
sity be  as  many  decimal  places  in  the  quotient  as  the 
number  of  decimal  places  in  the  dividend  exceed  the  num- 
ber of  decimal  places  in  the  divisor. 

We  deduce  the  same  conclusion  from  the  following  con- 
siderations. 

In  the  multiplication  of  decimals,  the  number  of  decimal 
places  in  the  product  equals  the  number  of  decimal  places 
in  both  factors.  In  the  division  of  decimals,  the  divisor' 
and  quotient  are  multiplied  together  to  produce  the  divi- 
dend ;  therefore,  there  must  be  as  many  decimal  places  in 
the  quotient  as  those  in  the  dividend  exceed  those  in  the 
divisor. 

The  pupil  should  bear  in  mind  that  he  can  affix  ciphers 
to  the  dividend  without  changing  its  value  ;  and  when 
necessary,  he  should  prefix  ciphers  to  the  quotient. 

1.  Divide  .0016016  by  1.12. 


134 


DECIMAL   FRACTIONS. 


[chap. 


TI 


OPERATION.  Explanation. — As  the  number 

1-12)'0016016(00143  Ans.  o^  Peaces  in  the  quotient  was  not 

12^2  equal  to  the  number  of  decimal 

places  in  the  dividend  minus  the 

481  number  of  decimal  places  in  the 

448  divisor,  so  the  two  ciphers  were 

— —  prefixed  that  the  required  number 

^•^"  of  decimal  places  could  be  cut  off. 

336  ^ 


2.  Divide  -00144  by  1-2. 

3.  Divide  'OOOOOtS  by  "005. 

4.  Divide  86-4  by  '24. 

5.  Divide  59-74514  by  13-6. 

6.  Divide  -001728  by  4-8. 

1.  Divide  2549052  by  24-6. 

8.  Divide  2448  by  '012 

Art.  152.  A  decimal  may  be  divided  by  10,  100, 1000, 
&c.,  by  removing  the  decimal  poiat  as  many  places  to  the 
left  as  there  are  ciphers  ia  the  divisor.     If  necessary,  pre- 
fix ciphers  to  the  dividend. 
r4-36 
Divide  \  ^^I'^ll  J>  by  10. 
431-2 


Divide 


Divide 


146-34 
3  24 
36  741 
14683 
47632-1 
1478*3 
231-46 
76-041 
31046-1 


by  100. 


> by  1000, 


PRACTICAL    QUESTIONS. 

1.  If  128  barrels  of  flour  be  worth  $784,  what  is  the 
value  of  I  barrel  ? 


ART.   152.]  PRACTICAL    QUESTIONS.  135 

2.  If  54  acres  of  land  cost  $816'75,  how  much  is  that 
an  acre  ? 

3.  What  cost  1  yard  of  broadcloth,  if  46,.  yards  cost 
$263-12? 

4.  What  cost  1  horse,  if  34  horses  cost  $4662-1  ? 

5.  What  cost  1  bushel  of  apples,  if  70  bushels  cost 
$43-75? 

6.  If  137  bushels  of  onions  cost  $154-12^,  what  will 
1  bushel  cost  ? 

7.  If  75  quarts  of  strawberries  cost  $4*6875,  what  will 
1  quart  cost  ? 

8.  If  275  bushels  of  corn  cost  $171*87^,  how  much  will 
1  bushel  cost  ? 

->• 

PRACTICAL    QUESTIONS    IN    DECIMALS    AND    THE    UNITED    STATES 
CURRENCY. 

1.  What  cost  8640  brick,  at  $4-25  a  1000  ? 
Solution If  1000  brick  cost  $4-25,  1  brick  will  cost  one 

thousandth  of  $425,  which  is  $-00425,  and  8640  brick  will  cost 
8640  times  $00425,  which  is  $36-72. 

2.  What  will  be  the  cost  of  4832  feet  of  boards,  at 
$6-50  a  1000? 

3.  What  will  be  the  cost  of  28460  feet  of  lumber,  at 
$2-18f  a  hundred  ? 

4.  What  cost  17640  feet  of  timber,  at  $9-45  a  1000  ? 

5.  What  cost  586  feet  of  pine  boards,  at  $25-12^  a 
1000? 

6.  What  must  be  paid  for  planing  46324  feet  of  boards, 
at  $1-45  a  1000? 

7.  What  is  the  value  of  14672  feet  of  hemlock  boards, 
at$6-37ia  1000? 

8.  A  speculator  bought  500  acres  of  land  for  $98 7f; 
and  250  acres  for  $647f .  He  sold  478^  acres  for  $1245^. 
How  much  land  has  he  remaining,  and  for  what  must 
he  sell  it  per  acre,  so  as  to  neither  gain  nor  lose  by  the 
operation  ? 

9.  Having  deposited  in  a  bank  $186050;  I  drew  out 
at  one  time  $84*87^;  at  another,  $47-12^;  at  another, 


136  DECIMAL    FRACTIONS.  [cHAP.    VI 

$485-18f ;  and  at  another,  $14t-31i.     How  much  have 
I  remaining  ia  the  bank  ? 

10.  A  gentleman,  who  was  on  a  journey  of  24 1^  miles, 
traveled  4  days,  at  the  rate  of  42|  miles  a  day;  what 
distance  still  remains  to  be  traveled  ? 

11.  Bought  a  house  and  lot  for  $324050,  and  paid  for 
improvements  on  the  same  $685"8T^.  I  then  sold  the 
property  for  $4985-621.  How  much  did  I  gain  by  the 
transaction  ? 

12.  A  land  dealer  has  in  one  farm  195* 7 5  acres;  in 
another  465f  acres;  in  another  483f  acres.  He  sold  t5^ 
acres  from  each.     How  many  acres  has  he  left  ? 

13.  A  drover  bought  cattle,  for  $n5'84  ;  mules  for 
$286-95  ;  horses,  for  $384-87| ;  and  sold  them  all  for 
$1847 -12i.     How  much  did  he  gain  by  the  speculation  ? 

14.  A  merchant  bought  cloth  for  $246-84  ;  silks  for 
$-387-8U;  and  sugar  for  $865-18f.  He  sold  the  cloth  at 
a  profit  of  $98-75;  the  silks,  at  a  loss  of  $104-121;  and 
the  sugar,  at  a  profit  of  $146'18f.  Did  he  gain  or  lose^ 
and  how  much  ? 

15.  A  merchant  bought  47-5  yards  of  cloth,  at  $4*75  a 
yd.;  and  sold  it,  at  $6'12i  a  yard.    How  much  did  he  gain  ! 

16.  Bought  285  sheep,  at  $2-12i  each;  and  sold  them 
for  25  young  cattle.  For  what  must  I  sell  the  cattle 
a  head  so  as  to  make  $75  by  the  operation  ? 

17.  How  much  money  must  be  paid  for  4-5  cwt.  of  ham, 
at  $14-25  a  cwt.;  14  barrels  of  flour,  at  $5-37i  a  barrel; 
8  barrels  of  fish,  at  $9 '621  a  barrel;  and  5f  barrels  of 
sugar,  at  $19'30  a  barrel  ? 

18.  What  sum  of  money  should  be  paid  for  75-75  pounds 
of  sugar,  at  $-1125  a  pound;  14  lbs.  of  tea,  at  $r37i  a 
pound;  15  lbs.  of  chocolate,  at  $-125  a  pound;  -and  5-75 
gallons  of  molasses,  at  $-37|  a  gallon  ? 

19.  A  speculator  bought  147i  acres  of  land,  at  $27-121 
an  acre;  and  232f  acres,  at$35f  an  acre.  He  sold  the 
first  tract,  at  $32181  an  acre;  and  the  second,  at  $28-371 
an  acre.  Did  he  gain  or  lose  by  the  operation,  and  how 
much  ? 

20.  Bought  4  pieces  of  cloth,  each  contaming  47|  yards, 


ART.    162.]  PRACTICAL   QUESTIONS.  13f 

for  $863-25;  of  which  25f  yards  have  been  sold,  at  $6-87^ 
a  yard.  What  will  be  the  gain  or  loss  on  the  whole,  if 
the  remainder  be  sold,  at  $5*95  a  yard  ? 

21.  A  drover  bought  247  cattle,  at  $25-87^  each.  He 
sold  84  of  them,  at  |32t5  each;  45  of  them,  at  $2245 
each;  and  the  remainder,  at  $28*12^  each.  How  much 
did  he  gain  by  the  speculation  ? 

22.  A  merchant  barters  to  a  farmer,  18*75  yards  of 
broadcloth,  at  $7*12^  a  yard;  47^  yards  of  muslin,  at 
$09|  a  yard;  6  pair  of  boots,  at  $4-37i  a  pair;— for  47 
bushels  of  corn,  at  $-57  a  bushel;  65f  bushels  of  wheat, 
at  $1-121  a  bushel.  The  difference  in  the  value  of  the 
articles  exchanged,  is  to  be  paid  in  money.  Which  of 
them  must  receive  money,  and  how  much  ? 

23.  A  merchant  bought  1246  bushels  of  wheat,  at 
$137i  a  bushel;  of  which  he  sold  to  one  man  463  bush- 
els, at  $1-45  a  bushel;  to  another  384^  bushels,  at  $1*87^ 
a  bushel.  At  what  price  per  bushel  must  the  remainder 
be  sold  so  as  to  gain  on  the  whole,  at  the  rate  of  $56  on 
a  1000"  bushels  ? 

24.  A  person,  having  $46*87^  was  desirous  of  purchas- 
ing an  equal  number  of  pounds  of  tea,  coffee,  and  sugar; 
the  tea,  at  $1'12J-  a  pound;  the  coffee,  $'62i;  and  the 
sugar,  $"12^  a  pound.  How  many  pounds  of  each  could 
he  buy  ? 

25.  Find  the  amount  of  a  store-bill  for  15f  yards  of 
cloth,  at  $3-371  a  yard;  20^  yards  of  silk,  at  $l-18f  a 
yard;  and  15  skeins  of  thread,  at  $-06i  a  skein. 

26.  Bought  16  barrels  of  sugar  for  $425-25,  and  sold 
the  same  at  a  profit  of  $1'87|  a  barrel.  At  what  price 
per  barrel  was  it  sold,  and  what  was  the  entire  profit  ? 

27.  A  merchant  bought  35  pieces  of  broadcloth,  each 
containing  18f  yards,  at  $6-18f  a  yard;  and  sold  it  so  as  to 
clear,  after  deducting  $4-37i  for  his  trouble,  $89'87i. 
At  what  price  per  yard  was  the  cloth  sold  ? 

28.  What  is  the  value  of  sugar  a  cwt.  when  '75  cwt. 
cost  $6-375;  and  what  should  be  paid  for  16f  cwt.  of 
sugar,  at  the  same  rate  ? 

29.  A  merchant  bought  of  one  farmer  225|  bushels  of 


138  DECIMAL    FRACTIONS.  [cHAP.    VI. 

wheat,  and  of  another  106|-  bushels,  at  $l'18f  a  bushel. 
He  made  195  bushels  of-  it  into  flour;  and  sold  the  flour, 
at  a  profit  of  $125*87|.  Will  he  gain  or  lose,  if  he  sells 
the  remainder  of  the  wheat,  at  $*93f  a  bushel. 

30.  A  drover  bought  146  cows,  at  $2T-18f  a  head; 
and  166  sheep,  at  $1*8T|-  each.  He  sold  83  of  the  cows, 
at  $28-12^-  a  head;  and" all  of  the  sheep,  at  ll'Sl^  each. 
At  what  rate  per  head  must  he  sell  the  remainder  of  his 
cows  so  as  to  make  a  profit  of  $125"93^  on  the  whole  ? 


Art.    153.   Reduction   op   Denominate   Numbers  to 
Decimals. 

1.  Reduce  15^.  9d.  3  far.  to  the  decimal  of  a  pound. 

Explanation. — We  annex  two  ci- 
phers to  the  3  far.,  which  reduces  it 
to  hundredths.  4  far.  make  1  penny  ; 
therefore,    ^  of  the   number  of  far- 

things   will    equal    the   number    of 

15'8125  s.  pence,    which    is   'lod.     This  being 

annexed  to  the  9d.  =  9'15d.      VVe 


•790625  of  a  pound,  next  divide  this  by  12,  to  reduce  it 
to  the  decimal  of  a  shilling,  and 
obtain  -81255. ;  which,  being  annexed  to  the  156\  gives  15  8125s. 
We  now  divide  this  by  20,  to  reduce  it  to  the  decimal  of  a 
pound,  and  obtain  •790625  of  a  pound  for  the  answer. 

2.  Reduce  ISs.  9d.  2  far.  to  the  decimal  of  a  pound, 
sterling. 

3.  Reduce  1  ft.  8  inches,  to  the  decimal  of  a  yard. 

4.  Reduce  1002-15  pwt.  9  grs.,   to  the  decimal  of   a 
poiind  Troy. 

5.  Reduce  15  cwt.  3  grs.  15*45  lbs.,  to  the  decimal  of 
a  ton. 

6.  Reduce  5  fur.  25  rds.,  to  the  decimal  of  a  mile. 
T.  Reduce  2  R.  25*5  P.,  to  the  decimal  of  an  acre. 

8.  Reduce  6  fur.  15  rds.  3  yds.  2  ft.  10  in.,  to  the  deci- 
mal of  a  mile. 

9.  Reduce  iE4  155.  lOd.  1  farthings,  to  the  decimal  of  a 
pound. 


ART.   154.]       REDUCTION    OF    DENOMINATE    DECIMALS.  139 

10.  Reduce  6  T.  12  cwt.  2  qrs.  14  lbs.  10  oz.  8  dr.  to 
the  decimal  of  a  ton. 


Art.  154.    Reduction  of  Denominate  Decimals,  to 

Whole  Numbers  of  a  lower  denominations. 

1.  Reduce  '735  of  a  pound,  to  shillings,  pence  and  far- 
things. 

OPERATION.  Explanation. — I  wish  to  reduce  '735  of  a 

£  '735  pound  to  shillings.     There  are  205.    in  £1 ; 

20  therefore,  20  times  the  number  of  pounds  = 

the  number  of  shillings,   20   X  -735  =  14- 75. 

14-700  s.  In  -75.,  how  many  pence  ?     There  are  12c/.  in 

12  Is. ;  therefore,   12  times  the  number  of  shil- 
lings equal  the  number  of  pence.     12  X    "7 


8-400  d.  _  8-4^      In  .4^?.  how  many  farthings  1  There 

^  are  4  farthings  in  1  penny ;  therefore,  4  time^ 

T~af\r\  f  -.  thenumber  of  pence  equal  the  number   of  far- 

i-DUU  lar.  ^^.^^^  ^  ^  .^^  ^  -^.g  ^^^    Therefore,  XO-735 

-=  145.  Sd.  1-6  far. 

2.  What  is  the  value  of  "389  of  a  pound  sterling  ? 

3.  What  is  the  value  of  '635  of  a  yard  ? 

4.  What  is  the  value  of  -451  of  an  ell  French  ? 

5.  What  is  the  value  of  '832  of  an  ell  English  ? 

6.  What  is  the  value  of  '^Sf  of  a  mile  ? 
1.  What  is  the  value  of  '895  of  an  acre  ? 

8.  What  is  the  value  of   975625  of  a  pound  Troy  ? 

9.  What  is  the  value  of  '875  of  a  score  ? 

10.  What  is  the  value  of  -95625  of  a  ream  of  paper  ? 

11.  What  is  the  value  of  '854  of  a  firkin  of  butter  ? 

12.  What  is  the  value  of  -7575  of  a  great  gross  ? 

13.  What  is  the  value  of -123  of  a  pound  sterling  ? 

14.  What  is  the  value  of  142857  of  a  bushel  of  salt  ? 
15  What  is  the  value  of  "783  of  a  bushel  of  wheat? 

16.  What  is  the  value  of  -857142857  of  a  bushel  of 
corn  or  rye  ? 

17.  What  is  the  value  of  -083  of  a  pound  sterling? 

18.  What  is  the  value  of  "16  of  a  cwt.  ? 


140  DECIMAL   FRACTIONS.  [cHAP.    n. 

19.  What  is  the  value  of  -123  of  a  mile  ? 

20.  What  is  the  value  of  '463  of  a  ton  ? 

PRACTICAL    QUESTIONS. 

1.  What  is  the  value  of  3  cwt.  2  qrs.  15  lbs.  of  sugar 
at  $5-^5  a  cwt.  ? 

2.  What  is  the  value  of  15  gallons,  3  qt.  1  pt.  of  molas 
ses,  at  $'87|  a  gallon  ? 

3.  What  is  the  value  of  16  bushels,  2  pks.  t  qts.  of  rye 
at  $1-3H  a  bushel? 

4.  What  is  the  value  of  84  yds.  3  qrs,  3  nas.  of  broad- 
cloth, at  $5-8ti  a  yard  ? 

5.  What  is  the  value  of  16  cwt.  2  qrs.  14'5  lbs.  of 
pork,  at  $14-93f  a  cwt.  ? 

6.  What  is  the  value  of  84  T.  14  cwt.  2  qrs.  15  lbs.  of 
hay,  at  $14-18f  a  ton  ? 

t.  What  is  the  value  of  34  lbs.  8^  oz.  of  butter,  at 
$'18f  a  pound  ? 

8.  What  will  it  cost  to  construct  14  miles,  5  fur.  25  rds. 
of  plank  road,  at  $1437-621  per  mile  ? 

9.  A  farmer  sold  34  bush.  3  pks.  *I  qts.  of  clover-seed, 
at  $6'84-i-  a  bushel,  and  in  payment  received  40  bushels 
2  pks.  1  pt.  of  grass-seed,  at$3'8t^a  bushel.  How  much 
remains  due  ? 

10.  A  tailor  paid  $1468-75  for  385  yds.  3  qrs.  3  nas.  of 
cloth;  I  of  which  he  sold,  at  $4*37^  a  yard;  and  the 
remainder,  at  $5-93f  a  yard.  How  much  did  he  gain  by 
the  bargain  ? 

11.  If  f  of  a  ton  of  hay  cost  $8-87i,  what  will  4  T. 
15  cwt.  3  qrs.  cost  ? 

12.  A  merchant  bought  125  hhds.  30-5  gals.  3  qts,  of 
molasses  for  $1585-12^;  and  sold  |  of  it  for  $21-75  a 
hogshead;  and  the  remainder,  at  $28-93|  a  hogshead. 
How  much  did  he  gain  by  the  operation  ? 

Reduction  of  Denominate  Fractions. 
Art.  155.  A  Denominate  fraction  is  a  fraction  of  any 
denominate  number;  as  |  of  a  yard,  |  of  a  mile,  &c. 


ART.    156.]  REDUCTION    OF    FRACTIONS.  141 

Reduction  of  denominate  fractions  is  changing  them  from 
one  denomination  to  another  without  altering  their  value. 

1.  Reduce  jf  g  of  a  gallon  to  the  fraction  of  a  gill. 

OPERATION   BY   CANCELLATION, 
gal. 

:r:r-x  V  -  V  -  V  -  =  —  of  a  gill. 
$00  ^  1  ^  1  '^  1       28  ^ 

xx% 

28 

Explanation There  are  4  quarts  in  1  gallon ;  therefore, 

4  times  the  number  of  gallons  equal  the  number  of  quarts. 
^I?  X  4  =  2  If  of  a  quart ;  (which,  for  convenience,  may  be  read 
in  the  form  of  a  compound  fraction.  There  are  2  pints  in  1 
quart ;  therefore,  twice  the  number  of  quarts  equal  the  number 
of  pints.  ^lB^XtXf  =  TT2ofa  pint.  There  are  4  gills  in  1 
pint ;  therefore,  4  times  the  number  of  pints  equal  the  number 
of  gills.  ^%^  X  t  X  f  X  f ,  equals  the  number  of  gills,  which, 
when  cancelled,  becomes  ^^  of  a  gill., 

2.  Reduce  -^\-^  of  a  pound  to  the  fraction  of  a  farthing. 

3.  What  part  of  a  grain  is  gsio  o  o^  ^  pound  Troy  ? 

4.  What  part  of  a  pint  is  -^^-^-^  of  a  bushel  ? 
6.  What  part  of  a  pound  is  y/o^  of  a  ton  ? 

6.  What  part  of  a  second  is  y  03F80  of  a  day  ? 

t.  What  part  of  a  foot  is  y/j 0  o^  ^  furlong  ? 

8.  What  part  of  a  dram  is  2 04  jo  of  a  hundredweight  ? 

Art.  156.  Reduction  of  Fractions  of  a  Lower,  to 
THOSE  of  a  Higher  Denomination. 

1.  Reduce  -f  of  a  farthing  to  the  fraction  of  a  pound. 
operation  by  cancellation. 

far. 
0         111  1         r  -, 

iy_xy  —  :=  of  a  Donnd. 

^  X  4  A  ^^  X  20      1120         ^ 

2 

Explanation. — f  of  a  farthing  is  what  part  of  a  penny  ? 
4  farthings  make  1  penny  \  therefore,  |  of  the  number  of  far- 
things equals  the  number  of  pense.     By  a  similar  method  of 


142  DECIMAL    FRACTIONS.  [CHAP.    VI 

reasoning  we  find  yV  of  the  number  of  pence  equal  the  num- 
ber of  shillings;  and  ^'o  «f  the  number  of  shillings  equal  the 
number  of  pounds. 

2.  What  part  of  a  pound  Troy,  is  f  of  a  grain  ? 

3.  What  part  of  an  acre  is  1|  feet  ? 

4.  What  part  of  10  days  is  |  of  a  minute  *? 

5.  What  part  of  20  bushels  is  |  of  f  of  a  gill? 

6.  What  part  of  a  rod  is  ^^  of  2^  inches  ? 

7.  What  part  of  8  miles  is  f  of  a  rod  ] 

8.  What  part  of  a  yard  is  J  of  |  of  f  of  an  ell  French  1 

Art.  l9>7«  Reduction  of  Simple  or  Denominate  Num- 
bers, TO  the  Fractional  Part  of  another  Simple  ob 
Denominate  Number. 

1.  What  part  of  £1  is  10s.  Qd.  1  far.  1 

operation.  * 

105.  6d.  1  far.  =  505  far.     101       ^ 

=  —  part. 

=9(50  far.     192 

Solution. — 4  farthings  make  1  penny;  therefore  1  far- 
thing is  I  of  a  penny.  6|c^.  =  Y^-  12t/-  make  1  shilling, 
therefore  ^^  of  the  number  of  pence  equals  the  number  of 
shillings.  j\X^i=Us.  10^s.  =  ^^^s.  20^.  make  £1 ; 
therefore  -^^^  of  the  number  of  shillings  equals  the  number 

of  £.       2*5  X    4^   =i§2'^" 

2.  What  part  of  3  yds.  is  4  E.  Fr.  2  qrs.  ? 

3.  What  part  of  3  cwt.  3  qrs.  is  2cwt.  3  qrs.  15  lbs.  1 

4.  W^hat  part  of  3  A.  3  R.  32^1-  P.  is  2  A.  2  R.  30  P.  ? 

5.  What  part  of  3  feet  square  is  3  square  feet  ? 

Art.  158.  To  find  tee  value  of  a  Denominate  Frao 
WON,  in  Whole  Numbers,  of  a  Lower  Denomination. 

1.  What  is  the  value  of  4  of  a  pound  sterling  ? 

OPEKATION 


Ans. 


£ 

7)5 

20       12      4 
s.        d.     qr. 

0     0    0 

0 

14     3    14 

ART.   158.]      ADDITION  OF  DENOMINATE  FRACTIONS.  148 

Explanation. — T  wish  to  find  f  of  £1,  but  ^  of  £1  is  the 
same  as  I-  of  £5  ;  heuce,  I  find  \  of  £5. 

2.  What  is  the  value  of  |  of  a  shilling  ? 

3.  What  is  the  value  of  f  of  a  cwt.  ? 

4.  What  is  the  value  of  f  of  a  yard  ? 
6.  What  is  the  value  of  ff  of  a  day  ? 

6.  What  is  the  value  of  |  of  a  mile  ? 

7.  What  is  the  value  of  |i  of  a  hogshead  of  wine  ? 

8.  What  is  the  value  of  |  of  a  year  ? 

9.  What  is  the  value  of  -5  of  an  ell  French  ? 
10.  What  is  the  value  of  j\  of  a  ton  ? 

Addition  of  Denominate  Fractions. 

Art.  159.  We  have  learned  that  whole  numbers  of 
different  denominations  connot  be  added;  the  same  is  true 
of  fractions  of  different  denominations.  Hence,  we  first 
find  the  value  of  the  given  fractions  by  Art.  158;  then  add 
them  together. 

1.  Add  |i  of  a  pound  to  -f  of  a  shilling. 

OPERATION. 

1^  of  a  pound     =145.    Sd. 
f  of  a  shilling.  =  lOd.  l\  far. 

Ans.  15s.    6fZ.  1\  far. 

2.  Add  I  of  a  pound  to  j\  of  a  shilling. 

3.  Add  /o  of  a  cwt.  to  |  of  a  quarter. 

4.  Add  4  of  a  ton  to  \-^  of  a  cwt. 

5.  Add  f  of  a  mile  to  |  of  a  furlong. 

6.  Add  I  of  an  acre  to  f  of  a,  rood. 

*7.  Add  f  of  a  hogshead  to  |  of  a  gallon. 

8.  Add  together  -f  of  a  bush.,  |  of  a  peck,  and  |  of  a 
quarter. 

9.  Add  together  |  of  a  ton,  f  of  a  cwt.,  and  4  of  a  qr, 
10.  Add  together  |  of  a  month,  f  of  a  week,  and  -f  of 

a  day. 


144  DECIMAL    FRACTIONS.  [cHAP.    VI. 

Art.  160.  Subtraction  of  Denominate  Fractions. 

1.  From  |-  of  a  mile  subtract  |  of  a  furlong. 

OPERATION. 

40      6i      3      12 
fur.  rds.  yds.  ft.    in. 

^  of  a  mile  =  6    8    4    2    8 
i|  of  a  fur.  =     28    3    0    5| 

Ans.  5  20    1    2    2f 

2.  From  f  of  a  bushel  take  |f  of  a  peck. 

3.  From  -f  of  a  week  take  |  of  a  day. 

4.  From  4  of  25  yards  take  f  of  6  E.  French. 
6.  From  -fi  of  23  tons  take  4  of  18  cwt. 

6.  A  company  agree  to  construct  25  miles,  8  fur.  18  rds. 
of  road,  but  after  constructing  6  mi,  2  fur.  23  rds.  and  2  ft. 
more  than  |  of  it,  they  relinquish  the  job.  How  much 
remains  to  be  constructed  ? 

1.  A  merchant  bought  f  of  15  hhd.  42  gals,  of  molasses, 
and  sold  j  of  2  hhd.  53  gals,  of  it.  How  much  remained 
unsold  ? 

8.  A  merchant  bought  14  cwt.  3  qrs.  18  lbs.  of  sugar, 
and  sold  |  of  it,  lacking  4  cwt.  1  qr.  15  lbs.;  how  much 
remains  unsold  ? 


practical  questions. 

1.  What  is  the  value  of  4  of  15  yards  of  cloth,  at  $^'62^ 
a  yard  ? 

2.  What  is  the  value  of  f  of  3  bushels,  3  pks.  *7  qts.  of 
gooseberries,  at  $06^  a  quart  ? 

3.  What  cost  f  of  41  cords,  110  feet  of  wood,  at  $5-81^ 
a  cord  ? 

4.  What   cost  1   pound  of  tea,  if   11^  pounds   cost 
$13-826? 

5.  What  will  6  cwt.  3  qrs.  20  lbs.  of  honey  cost,  at 
$18'8tiacwt.  ? 

6.  What  will  14  bushels,  2  pks.  1  qts.  1  pt.  of  grass- 
Beed  cost,  at  $6-621  a  bushel  ? 

1.  If  it  require  4  hours  20  minutes  for  a  man  to  cut 


A.RT.  160.]  PRACTICAL   FRACTIONS.  145 

I  cord  of  wood,  how  many  days  of  8  hours  and  40  minutes 
«ach,  will  be  required  to  cut  84t  cords  84  feet  ?  ** 

8.  Four  persons  share  625  pounds  of  sugar  as  follows  : 
the  first  takes  i  of  |  of  the  whole;  the  second  takes  |  of 
3.  of  the  remainder;  the  third  takes  f  of  |f  of  what  now 
remains;  and  the  fourth  takes  what  is  left.  How  much 
did  each  receive  ? 

9.  A  received  |  of  a  certain  quantity  of  molasses;  B  |; 
C  I  of  the  remainder;  and  D  what  then  remained.  It 
now  appears  that  C  has  64  gals,  more  than  A  and  B 
together.     How  much  did  each  receive  ? 

10.  A  farmer,  owning  864  A.  3  R.  39  P.  of  land,  divided 
I  of  it  equally  among  4  of  his  sons.  How  much  did  each 
son  receive,  and  how  many  acres  had  the  father  remain- 
ing ? 

11.  Bought  184  gals.  3  qts.  of  molasses,  at  $'3*r|  a 
gallon,  and  used  2t  gals,  2  qts.  of  it;  how  must  I  sell  the 
remainder  per  gallon  so  as  to  receive  $3-84^  more  than 
the  whole  cost  ? 

12.  A  person  gave  |  of  all  his  money  for  a  horse;  -i-  of 
the  remainder  for  a  colt;  and  |  of  what  then  remained  for 
a  cow.  He  then  had  remaining  $8'8t|.  What  was  the 
cost  of  each,  and  how  much  money  had  he  at  first  ? 

13.  A  merchant  gave  for  some  raisins  }  of  all  his  money; 
for  some  cinnamon  ^  of  all  his  money;  for  some  sugar  |  of 
what  remained;  for  some  flour  |  of  what  then  remained; 
and  what  still  remained  he  gave  for  some  butter.  What 
did  each  article  cost  him,  providing  the  sugar  cost  $136*18f 
more  than  the  flour  ? 

14.  A  certain  sum  of  money  is  to  be  divided  among  4 
persons;  the  first  is  to  have  ^  of  it;  the  second  i  of  it; 
the  third  |  of  what  remains;  and  the  fourth  the  remainder. 
What  was  the  sum  to  be  divided,  and  how  much  did  each 
receive,  providing  the  third  received  $14*I'93f  less  than 
the  first  and  second  together? 

15.  How  much  butter  at,$-18f  a  pound,  must  be  given 
for  25  gals.  3  qts.  1  pt.  of  molasses,  at  $-3t^  a  gallon  ? 

16.  From  a  piece  of  cloth  containing  147  yds.  4  E. 
French,  three  suits  of  clothes,  each  requiring  6  E.  English, 


146  DUODECIMALS.  [CHAP.    VI. 

were  taken.     How  much  would  the  remainder  come  to,  at 
$5-18f  a  yard  ? 

It.  How  many  inches  in  f  of  an  E.  E.;  f  of  an  E.  Er.; 
and  f  of  a  quarter  ? 
J  18.  A  merchant  lost  from  a  hogshead  of  molasses  }  of 
it,  +  A  of  a  gallon  and  |  of  a  quart.  How  much  of  the 
hogshead,  expressed  deciifially,  leaked  out,  and  how  much 
remained  in  ? 

19.  Bought  15  tons  14  cwt.  3  qrs.  24  lbs.  of  iron,  and 
sold  10  tons  5  cwt.  1  qr.  15  lbs.  of  it.  What  is  the  value 
of  ^  of  what  remains,  at  $'06|  a  pound  ? 

20.  Bought  a  quantity  of  grain  for  $358"84  ;  and  sold 
\^  of  it  to  one  man;  f  of  the  remainder  to  another  man; 
and  used  |  of  the  remainder  myself.  What  is  the  value 
of  the  remainder  ? 

21.  A,  B,  C,  and  D  worked  together  on  this  condition: 
A  was  to  receive  $60*06  of  it,  and  y^  of  the  remainder; 
B  was  to  receive  $70'0t  and  j\  of  the  remainder;  C  was 
to  receive  $80'08  and  j\  of  the  remainder;  and  I)  took 
what  then  remained.  By  this  division  each  man  received 
the  same  sum.     How  much  did  their  wages  amount  to  ? 


DUODECIMALS. 

Abt.  161.  Duodecimals  are  a  kind  of  denominate 
numbers,  the  denominations  of  which  increase  uniformly 
in  a  twelve-fold  ratio.  Its  denominations  are  the  foot  (ft.), 
which  is  the  unit  ;  the  inch,  or  prime  ('),  J^  of  the  foot, 
the  secoTid  ("),  y^2  of  the  prime  ;  the  third  ('"),  ^\  of  the 
second  ;  and  so  on,  indefinitely.  The  accents  that  distin- 
guish the  denominations  below  feet,  are  called  Indices. 

Duodecimals  are  applied  to  the  measurement  of  surfaces 
and  solids. 

TABLE. 

12  Fourths  ("")  make  1  X^ird,            marked  '" 

12  Thirds  '*  1  Second, 

12  Seconds  "  1  Prime,  or  Inch,  " 

12  Primes,  or  Inches  '*  1  Foot,                 "      ft 


art.  163.]       multiplication  of  duodecimals.  14 1 

Addition  axd  Subtraction  of  Duodecimals. 

Art.   162.  Duodecimals  are  added  and  subtracted  the 
same  as  other  Denominate  numbers. 

1.  Add  together  6  ft.  4'  5"  8'",  8  ft.  4'  8"  9'",  1,  ft. 
3'  8"  9'",  and  12  ft.  9'   11"  10'". 

2.  What  is  the  sum  of  1^  ft.  8'  9"  11"',  14  ft.  6'  t", 
8  ft.  9'  11"  4"',  and  16  ft.  9'  10"  11"'  ? 

3.  What  is  the  sum  of  20  ft.  9'  11"  6'"  1"",  14  ft.  8' 
9"  10'",  12  ft.  9'  8"  10'"  8"",  8  ft.  11"",  and  6  ft.  9'  ? 

4.  From  84  ft.  8'  9"  11'"  3'"',  subtract  66  ft.  11'  8' 
4'"  9"". 

5.  What  is  the  sum,  and  what  is  the  diflference  of  84 
ft.  3'  8"  9'"  2"",  and  48  ft.  9'  T  IV"  10"". 

6.  What  ifi  the  sum,  and  what  is  the  diflference  of  13t 
ft.  3'  9"  4'"  6""  and  98  ft.  9'  10"  11"'  t"". 


Multiplication  of  Duodecimals. 

In  Duodecimals,  the  foot,  when  used  to  express  surfaces, 
contains  144  sq.  in.,  and  when  used  to  express  solids, 
1728  cu.  in.  Consequently,  in  the  measurement  of  sur- 
faces, 5'  would  equal  ^^  of  a  square  foot,  instead  of  a  linear 
foot ;  that  is,  j^2Xl44  sq.  in. =60  sq.  in.  In  the  measure- 
ment of  solids,  5'  would  eq^ual  ^^g  of  1728  cu.  in.  (a  cubic 
foot)  =720  cu.  in. 

From  the  .preceding  remark  we  infer  that  a  strip  of  sur- 
face 1  inch  wide  and  12  inches  long,  makes  1'  square 
measure;  and  that  a  slab  1  inch  thick,  12  inches  long,  ai«i 
12  inches  wide,  makes  1'  solid  measure. 

1.  What  is  the  product  of  8  ft.  5'  by  9  ft.  r  ? 

OPERATION.         Explanation. — 5  =:  -^\,  and  7'  =  -^-^  of  a 

8  ft.  5'       foot-     Therefore,  we  say^  7'  X  5'  =  f^\  of 

9  ft.*  7'      ^  foot,  which  is  35"  =  2'  11";    we  write 
down  the  11"  and  carry  the  2'  to  the  next 


4  ft.  10'     11"      product.     7'  X  8  ft.  =  f  |  of  a  foot,  which 
75  ft*    9'  is  56',  and  2'  added  =  58',  which  equals 

'. 4  ft.  10',  this  we  write  down.     9  ft.  X  5' 

80  ft.   7'    11"      =  tl  of  ^  foot,  which  is  45'  =  3  ft.  9' ; 
write  down  the  9'  and  carry  the  3  ft.  to  the 


148  DUODECIMALS.  ^CHAP.    VI 

next  product.  9  ft.  x  8  ft.  =  72  ft.  and  3  ft.  added  =  75  ft. 
The  sum  of  these  partial  products  gives  the  required  product, 
which  is  80  ft.  7' 11". 

Remakk. — It  has  already  been  stated  that  it  was  impossible  to  multiply  one 
concrete  number  by  another.  I'he  above  example  may  appear  at  first 
thought  to  be  contrary  to  that  statement,  but«it  must  be  remembered  that  the 
multiplier  is  considered  an  abstract  number. 

2.  What  is  the  product  of  14  ft.  T  2"  by  6  ft.  3'  5"  ? 

3.  Wtiat  is  the  area  of  a  marble  slab,  the  length  of 
which  is  9  ft.  8'  11",  and  width  3  ft.  1'  ? 

4.  How  many  square  feet  are  contained  in  the  floor  of 
a  room  40  ft.  10'  long,  32  ft.  8'  wide  ? 

6.  How  many  square  feet  in  10  boards,  each  18  ft.  10 
long  and  1  ft.  8'  wide  ? 

6.  How  many  square  feet  of  boards  will  it  take  to 
inclose  a  piece  of  land  80  ft.  10  in.  long,  and  60  ft.  8  in. 
wide,  with  a  close  fence  T  ft.  6  in.  high  ? 

7.  How  many  square  yards  in  a  floor  which  is  48  ft.  6' 
long,  and  36  ft.  10'  wide  ? 

8.  What  will  the  plastering  of  a  room  cost,  at  18  cents 
a  square  yard,  the  length  of  which  is  30  ft.  10  in.,  width 
24  ft.  6  in.,  and  height  of  ceiling  8  ft.  4'  ? 

9.  In  a  certain  building  there  are  32  windows;  in  each 
window  16  lights;  and  each  light  is  1  ft.  10'  by  11'.  How 
many  square  feet  of  glass,  in  the  32  windows  ? 

10.  In  a  certain  room  24  ft.  long,  18  ft.  6'  wide,  and 
10  ft.  2'  high,  there  are  6  windows,  each  6  ft.  2'  long,  3  ft. 
10'  wide;  and  3  doors,  each  6  ft.  10'  by  3  ft.  What  will 
be  the  cost  of  plastering  this  room,  at  16  cents  a  square 
yard  ? 

11.  How  many  solid  feet  in  a  pile  of  wood  24  ft.  6  in. 
long,  6  ft.  5'  high,  and  4  ft.  6'  wide  ? 

Remark.— Multiply  the  length,  height,  and  width  together,  to  find  the  solid 
contents. 

12.  How  many  cubic  feet  in  a  stick  of  timber  32  ft.  9' 
long,  2  ft.  2'  wide,  an'd  2  ft.  8'  thick  ? 

13.  How  many  bricks,  each  8  in.  long,  4  in.  wide,  and 
2  in.  thick,  are  required  to  build  a  wall  144  feet  long,  6 
ft.  6  in.  high^  and  three  bricks  wide,  no  allowance  being 
made  for  the  mortar  ? 


art.  164.]  division  of  duodecimals.  149 

Art.  164.  Division  of  Duodecimals. 

1.  There  are  8  ft.  5'  3"  in  the  surface  of  a  marble  slab, 
the  length  of  which  is  3  ft.  9'  ;  what  is  its  width  ? 

OPERATION.  Explanation. — 3  ft.  is  con- 

3  ft.  9')8  ft.  5'  3"(2  ft.  3'  Ans.     tained  in  8  ft.  2  times.     Mul- 

7  f^  Q'  tiplying  the  whole  divisor  by 

L 2  ft.   give  7  ft.  6'  for  the  pro- 

21'  3"  duct,  whiclr~we  subtract  from 

11'  3'/  the    corresponding    denomina- 

tions  of  the  dividend,  and  ob- 


0  tain    11'   for  a   remainder,    to 

which  annex  the  next  denomi- 
nation of  the  dividend,  and  we  have  11'  3".  3  ft.  is  contained 
in  11',  3'  times.  The  divisor  being  multiplied  by  this  3'  give 
11'  3",  which  being  subtracted  from  the  last  remainder  leaves 
nothing.     Therefore,  the  marble  slab  was  2  ft.  3'  in  width. 

Remark. — If  the  student  will  bear  in  mind  that  the  superficial  contents  oi 
any  surface  is  found  by  multiplying  the  length  by  the  breadth,  he  will  readily 
understand  that  dividing  the  superficial  contents  of  any  surface  by  its  length 
will  give  its  width,  or  by  its  width  will  give  its  length.  Also,  since  the  so- 
lidity of  a  body  is  found  by  multiplying  its  three  dimensions  together,  if  we 
divide  its  cubical  contents  by  the  product  of  either  two  of  its  dimensions,  the 
quotient  will  be  the  other  dimension. 

The  number  of  indices  to  be  annexed  to  any  term  of  the  quotient  can  be 
readily  determined,  since  the  indicts  of  the  quotient  added  to  the  indices  of  the 
divisor  must  equal  those  of  the  dividend. 

2.  There  are  489  sq.  ft.  8'  0"  2'"  1!'",  in  the  surface 
of  a  floor.  The  length  of  the  floor  is  87  ft.  1'  11".  What 
is  its  width  ? 

3.  There  are  28  sq.  ft.  3'  11"  2"',  in  the  surface  of  a 
table;  the  length  of  which  is  6  ft.  9'  T" ;  what  is  its  width  ? 

4.  The  area  of  a  certain  pond,  the  length  of  which  is 
43  ft.  9'  6",  is  1075  sq.  ft.  0'  3"  0"'  6"".  What  is  its 
width  ? 

5.  A  stick  of  timber  is  3  ft.  2'  wide,  2  ft.  11'  thick,  and 
contains  135  cu.  ft.  10'  2"  1"'.     What  is  its  length  ? 

6.  The  area  of  a  pond  is  3978  ft.  1'  6";  its  length  is 
100  ft.  6'.     What  is  its  width? 

7.  The  area  of  a  marble  slab  is  27  ft.  0'  7"  9'"  6"";  its 
length  is  7  ft.  6'  3".     What  is  its  width  ? 

8.  The  area  of  a  hall  is  103  ft.  4'  5"  8"'  4"";  its  width 
is  6  ft.  ir  8".     What  is  its  length  ? 


1.60  REDUCTION   OF   CURRENCIES.  fcHAP.   VI. 


REDUCTION  OF  CURRENCIES. 

Art.  165.  Reduction  of  Currencies  teaches  how  to 
find  the  value  of  the  denominations  of  one  currency  in  the 
denominations  of  another. 

The  value  of  a  dollar,  expressed  in  shillings  and  pence, 
is  not  the  same  in  different  States  of  the  Union,  and  in 
different  countries.  This  difference  may  be  learned  from 
the  following 

TABLE. 

I  North  Carolina,  i  """"^y- 

r  New  England  States, "] 
jg-.  .     I  Virginia,  (  =  6s.  =  £j\,  called  New  Eng- 

^         ]  Kentucky,  j  land  currency. 

[  Tennessee,  j 

f  New  Jersey,  ^ 

^-.  .     I  Pennsylvania,  1  =  75.  6d.  =  £|,  called  Penn- 

*    ^°  I  t)elaware,  j  sylvania  currency. 

i  Maryland,  J 

^1  .     j  South  Carolina,  )  4s.  Sd.  =  £/„,  called  Georgia 

^    ^^  i  Georgia,  j  currency, 

^-j  .     (  Canada,  )  5s.  =  £|,  called  Canada  cur- 

^^  ^°  j  Nova  Scotia,  [  rency. 

The  legal  value  of  £1  English  or  Sterling  money,  is  $4.84, 
as  fixed  by  an  act  of  Congress  in  1842. 

The  above  Table  gives  the  value  of  $1,  expressed  in  the 
fraction  of  a  pound,  in  the  different  currencies.  The  value 
of  £1  in  each  of  the  above  currencies  is  found  by  analysis, 
thus,— If  £|  =  $1,  £}  =z  $i  and  £^  or  £1  =  5  times  i, 
which  is  $f .  In  a  similar  manner  from  the  above  table, 
we  can  form  the  following 

TABLE. 

£1  =  $f ,    New  York  currency. 
£1  =  $y,  New  England  currency. 
£1  =  $f ,    Pennsylvania  currency. 


£1  =  $Y,  Georgia  currency. 
£1  =  |4;    Canada  currency. 


ART.  167.]  REDUCTION    OF   CURRENCIES.  151 

1.  Redace  $321*75  to  its  equivalent  value  in  Penn- 
sylvania currency. 

OPERATION. 

$321-75  X  i  =  £120-65625, 
which  equals  £120  135.  Id.  2  far. 

2.  Reduce  $345-25  to  its  equivalent  value  in  New  York 
currency. 

3.  Reduce  $684"12|-  to  its  equivalent  value  in  New 
England  currency. 

4.  Reduce  $67*84  to  its  equivalent  value  in  Georgia 
currency. 

5.  Reduce  $846'87|  to  its  equivalent  value,  Canada 
currency. 

6.  Reduce  $846*625  to  its  equivalent  value  in  English 
or  Sterling  money. 

Art.  166.  Reduction  of  Pounds,  Shillings,  &c.,  op 
different  currencies,  to  federal  money. 

1.  Reduce  £75  15^.  Qd.  New  York  currency,  to  Federal 
money. 

OPERATION. 

£75  15.S.  6d.  =  £75-775. 
£75-775  X  I  =  $189-4375. 

2.  Reduce  £154  10^.  8^.  New  England  currency,  to 
Federal  money. 

3.  Reduce  £346  I65.   9d.    Pennsylvania  currency,   to 
Federal  money. 

4.  Reduce  £843  15^.  8^.  Georgia  currency,  to  Federal 
money. 

5.  Reduce  £49  ISs.  lid.  Canada  currency,  to  Federal 
money. 

6.  Reduce  £784  17^.  lOd.  Sterling,  to  Federal  money. 

Art.  167.   The  following  table  shows  the  value  of 
some  of  the  foreign  coins  at  their  standard  value  : — ■ 

1  Pound  Sterling,  or  Sovereign,      .            .            .  $4- 84 

1  Guinea,  English,       .            ,            .            .  .         500 

1  Crown, 1-06 


152 


ALIQUOT   PARTS, 


[chap.  VI. 


1  Shilling  piece,  English,  .  .  .  ,        -23 

1  Franc,  ......  -186 

1  Doubloon,  Mexico,        .  .  .  .     15  60 

1  Specie  Dollar  of  Sweden  and  Norway,       .  .  1-06 

1  Specie  Dollar  of  Denmark,      .  .  .  .105 

1  Thaler  of  Prussia  and  N.  States  of  Germany,        .  -69 

1  Florin  of  Austrian  Empire  and  City  of  Augsburg,     .       -485 
1  Ducat  of  Naples,  ....  -80 

1  Ounce  of  Sicily,  .....       2-40 

1  Pound  of  British  Provinces,  Nova  Scotia,  New  Bruns- 
wick, Newfoundland,  and  Canada,         .  .  4- 00 

Note. — A  little  reflection  will  enable  the  pupil  ta  reduce  any  of  these  foreign 
coins  to  Federal  Money,  or  Federal  Money  to  foreign  coins. 

Aliquot  Parts. 

Art.  168.  The  half,  third,  fourth,  fifth,  &c.,  of  any 
quantity,  is  an  Aliquot  Part  of  that  quantity. 

Art.  169.  Analysis  is  applied  to  arithmetical  solu- 
tions, when  the  various  factors  of  the  question  and  their 
relations  are  traced  out,  forming  a  process  of  reasoning. 


ANALYSIS   BY   ALIQUOT   PARTS. 

1.  What  is  the  value  of  4  cwt.  2  qrs.  12^  lbs.  of  sugar, 
at  $8.84  a  cwt  ? 


2  qrs.  =  \  cwt. 


121  lbs.  =  :^  of  2  qrs. 


OPERATION. 

$  8.84 
4 


35.36  value  of  4  cwt. 
4.42  value  of  2  qrs. 
1.101  value  of  121  lbs. 

So'ssi  Ans. 


2.  What  is  the  value  of  25  lbs.  5  oz  12  pwts.  of  silver 
ware,  at  $54*18f  a  pound  ? 

3.  What  is  the  value  of  6  tons  5  cwt  3  qrs.  of  iron,  at 
$35-371  a  ton  ? 

4.  What  is  the  value  of  16  cwt.  2  qrs  15  lbs.  of  sugar 
at  $9-371  a  cwt.  ? 


ART.  170.]  CANCELLATION.  153 

5.  What  is  the  yalue  of  346  bushels  3  pks.  1  qt.  of  rye, 
at  $-93f  a  bushel  ? 

6.  What  is  the  value  of  3  pks.  6  qts.  of  cherries,  at 
$1-12^  a  peck? 

t.  A  market  woman  bought  2  bushels  3  pks.  4  qts.  of 
strawberries,  at  $287  a  bushel.  How  much  did  she  pay 
for  them  ? 

8.  A  merchant  bought  25  yds.  2  qrs.  2  nas.  of  silk,  at 
$1'87|  a  yard ;  and  3t  yds.  3  qrs.  3  nas.  of  broadcloth,  at 
$4*95  a  yard.     What  did  the  whole  amount  to  ? 

9.  A  gentleman  bought  a  lot  of  land  containing  4*7  A. 
2  R.  25  P.,  at  $85*3T^  an  acre.  How  much  did  he  pay 
for  the  lot  ? 

Cancellation. 
Art.  170.  Cancellation,  in  arithmetic,  consists  in  re- 
jeciing  equal  factors  from  a  divisor  and  dividend,  which 
does  not  change  the  value  of  the  fraction  ;  it  being  the 
same  as  dividing  both  divisor  and  dividend  by  the  same 
number.     (See  Art.  116.    Proposition  6th.) 

ANALYSIS    BY    CANCELLATION. 

1.  If  I  of  a  yd.  of  cloth  cost  $f ,  what  will  |  of  a  yd.  cost  ? 

Analysis. — If  f  of  a  yard  cost  $|,  ^  of  a  yard  will  cost  |  of 
$f ;  and  §  (1  yard)  will  cost  ^  of  $f.  If  1  yard  cost  f  of  $|, 
I  of  a  yard  will  cost  |  of  i  of  $f  ]  and  |  of  a  yard  will  cost  | 
offoflf,  =  $f.  Ans. 

OPERATION. 

3 

Remark. — Those  who  prefer  can  place  the  numerators  of  the  fractions  on 
the  right  ol  a  perpendicular  line,  one  under  another  ;  and  the  denominators, 
in  a  similar  way.  on  the  left  of  the  same  line,  and  thereby  avoid  writing  the 
sign  01  multiplication.    Thus  : 


2 
3  0 


5 
$ 

$  Ans.  U. 


154  RATIO.  [chap.   VII. 

2.  If  I  of  a  yard  of  cloth  cost  $6,  what  will  f  of  a  yard 
cost? 

3.  How  much  will  f  of  a  ton  of  hay  cost,  when  4i  tons 
cost  $13-39? 

4.  Allowing  a  horse  to  travel  |  of  a  mile  in  4  minutes, 
what  distance  would  he  travel  in  48  minutes  ? 

5.  If  6  men  can  perform  a  certain  piece  of  work  in  24'6 
days,  in  what  time  can  24  men  perform  the  same  work  ? 

6.  A  gave  towards  the  building  of  a  church  $140,  which 
was  ^  as  much  as  B  gave,  and  B  gave  |  as  much  as  C. 
How  much  did  C  give  ? 

t.  If  31  bushels  of  corn  are  worth  2i  bushels  of  rye, 
how  many  bushels  of  corn  are  worth  14|  bushels  of  rye  ? 

8.  A  has  I  as  much  money  as  B;  and  f  as  much  as  C, 
who  has  ^  as  much  as  D,  who  has  $2400.  How  much 
have  A,  B,  and  C  respectively  ? 


CHAPTER  YII. 

RATIO. 


Art.  ITl.  Two  numbers  or  quantities  of  the  same 
denomination,  may  be  compared  together  in  two  ways, — 

First.  By  means  of  an  Arithmetical  Ratio,  which  is 
expressed  by  their  difference. 

Secondly.  By  means  of  a  Geometrical  Ratio,  which  is 
expressed  by  the  number  of  ti7nes  the  one  contains  the  other. 

The  word  Ratio,  when  used  alone,  refers  to  a  geometrical 
ratio. 

Ratio  is  the  relation  which  one  number,  or  quantity, 
bears  to  another  of  the  same  denomination,  and  is  expressed 
by  the  quotient  arising  from  dividing  the  first  by  the 
second,  or  by  dividing  the  second  by  the  first.   . 

When  we  speak  of  the  ratio  of  one  number  to  another 
in  Arithmetic,  we  shall  refer  to  the  quotient  arising  from 
dividing  the  second  terra  by  the  first,  as  the  first  term  iu 


ART.    172.]  PROPORTION,  155 

a  simple  proportion  is  made  the  divisor.  Thus,  the  tatio 
of  2  feet  to  8  feet  is  4,  or  expressed  in  the  form  of  a  frac- 
tion, is  f.  The  ratio  of  two  quantities  is  usually  expressed 
by,  (:)  being  placed  between  them;  thus,  2  :  8,  which 
equals  |  or  4. 

A  ratio  cannot  be  a  concrete  or  denominate  number; 
neither  is  there  a  ratio  between  quantities  of  diiferent 
denominations. 

1.  What  is  the  ratio  of  5  yards  to  25  yards  ? 

2^  What  is  the  ratio  of  4  inches  to  36  inches  ? 

3.  What  is  the  ratio  of  8  apples  to  72  apples  ? 

4.  What  is  the  ratio  of  24  sheep  to  96  sheep  ? 

5.  What  is  the  ratio  of  9  pounds  to  108  lbs.  ? 

6.  What  is  the  ratio  of  4  feet  to  $16  ? 

7.  What  is  the  ratio  of  4  sheep  to  24  horses  ? 


PROPORTION. 

Art.  172.  When  two  quantities  have  the  same  ratio 
as  two  other  quanities,  the  four  quantities  are  said  to  be 
in  Proportion.  Thus,  the  ratio  of  8  bushels  to  32  bushels, 
is  the  same  as  the  ratio  of  $3  to  $12. 

Proportion  is  an  equality  of  ratios  of  numbers  compared 
together,  two  and  two. 

Quantities  are  shown  to  be  in  proportion  by  means  of 
dots;  for  example,  the  above  proportion  is  written, 

bush.         bush.  $  $ 

8     :     32     ::     3     :     12 
And  is  read  8  busels  is  to  32  bushels,  as  $3  is  to  $12. 

Rkmark. — The  two  dots  placed  between  the  first  and  second,  also  between 
the  third  and  fourth  terms,  in  the  above  proportion,  are  contractions  of  the  sign 
of  division  (-;-),  the  horizontal  line  being  omitted.  The  four  dots  between  the 
second  and  third  are  contracted  from,  and  equivalent  to,  the  sign  of  equality. 
Hence,  the  above  was  formerly  written, 

8  bush.  -T-  32  bush.  :=  $2  -f-  $K.  This  expression  indicates  that  the  ratio  is 
found  by  dividing  the  first  term  by  the  second.  The  English  mathematicians 
have  adopted  this  method  of  expressing  the  ratio  of  one  number  to  another, 
while  the  French  divide  the  second  by  the  first,  as  previously  directed. 

The  first  two  terms  of  pr9portion  are  called  the  first 
couplet ;  the  second  two  terms,  the  second  couplet. 


156  SIMPLE   PROPORTION.  [cHAP.    VIL 

The  first  term  of  each  couplet  is  called  the  Anteadentj 
and  the  second  term  is  called  the  Consequent. 

The  Jlrst  and  fourth  terms  of  a  proportion  are  called 
the  Extremes,  and  the  second  and  third  terms  are  called  the 
Means. 

Since,  in  a  proportion,  the  quotient  obtained  by  dividing 
the  second  term  by  i\\Q  first,  is  equal  to  the  quotient  obtained, 
by  dividing  the  fourth  term  by  the  third,  we  can  readily 
deduce  the  following 

PROPOSITIONS. 

1.  The  product  of  the  mea7is  is  equal  to  the  product  of 
the  extremes.     Therefore, 

2.  If  the  product  of  the  means  le  divided  hy  one  extreme, 
the  quotient  will  be  the  other  extreme.     Or, 

3.  If  the  product  of  the  extremes  le  divided  hy  one  mean, 
the  quotient  will  be  the  ot/ier  mean. 

4.  The  fourth  term  of  a  proportio7i  is  equal  to  the  third 
term,  multiplied  by  the  ratio  of  the  first  term  to  the  second. 

Suggestion. — These  propositions  being  understood,  the  pu- 
pil can  readily  determine  the  remaining  term  of  a  proportion, 
if  any  three  of  them  be  given. 


SIMPLE    PROPORTION. 

Art.  IT'S,  Simple  Proportion  teaches  the  method  of 
finding  the  fourth  term  of  a  proportion,  by  knowing  the 
other  three. 

Art.  174.  In  stating  a  question  in  Simple  Proportion, 
the  FIRST  and  second  terms  must  be  of  the  same  kind  or 
denomination,  and  the  third  terra  like  the  answer  sought. 
If  the  answer  is  to  be  greater  than  the  third  term,  the 
larger  of  the  two  remaining  terms,  must  occupy  the  second 
place, — if  smaller,  the^^r^^  place.  Then  proceed  accord- 
ing to  Proposition  2nd,  or  4th. 

1.  If  12  bushels  of  wheat  cost  $21*60,  whax  will  29 
bushels  cost  1 


ART.  174  ]  SIMPLE    PROPORTION.  151 

Explanation — The  answer  sought  is  to  be  in  dollars,  there- 
fore, we  have  the  $2100,  for  the  titird  term.  The  answer  is  to 
be  greater  then  the  third  term,  because  29  bushels  will  cost 
more  then  12  bushels  :  hence,  we  have  the  larger  number,  29 
for  the  second  term  and  12  for  the  first.     Thus : 

bnsh.        bush.  $ 

12    :    29  : :  21-60 
29 


19440 
4320 


12)626-40,  the  product  of  the  means. 

^  $52-20,  the  other  extreme,  or  4th  term. 
The  above  question  can  as  well  be  solved,  by  finding  the  ratio, 
of  the  first  to  the  second  term.   Thus  :    (See  Prop.  4th). 

OPERATION. 

1-80 
29     ^/•00     ^ 
^X—^— =$52-20.  Ans. 

2.  What  will  24T  yards  of  cloth  cost,  if  25  yards  cost 
$144-60? 

3.  What  will  347  bushels  of  corn  cost,  if  84  bushels  cost 
$66-40  ? 

4.  What  will  384  bushels  of  wheat  cost,  if  35  bushels 
cost  $30-80? 

5.  What  will  341  boxes  of  raisins  cost,  if  312  boxej 
cost  $436-121  ? 

6.  If  a  man  travel  485  miles  in  18  days,  how  far  at  this 
rate  will  he  travel  in  125  days  ? 

T.  A  garrison  of  125  men  has  provisions  for  35  days. 
How  many  of  the  men  must  be  discharged,  that  the  re- 
mainder may  be  supported  for  125  days  ? 

8.  If  43  men  can  do  a  certain  piece  of  work  in  47^  days, 
how  many  days  will  it  take  15  men  to  do  the  same  ? 

9.  A  man  bought  cows,  at  $12315  for  8.  How  much 
at  the  same  rate  would  35  cows  cost. 

10.  Bought  23  pieces  of  delaine,  each  containing  41| 
yards,  at  the  rate  of  $24-45  for  45  yards.  How  much  did 
it  all  cost  ? 


158  PROPORTION,  .      [chap.    VII 

11.  If  a  company  of  190  men  consume  54  barrels  of 
flour  in  6  weeks,  how  many  barrels  would  it  take  to  last 
them  1  year  ? 

12.  If  $273  in  3  years  gives  $18-621  interest,  how  long 
will  it  require  to  give  $184  interest  ? 

13.  If  $83  in  two  years  8  months  give  $12*37^  interest, 
what  sum  in  the  same  time  will  give  $3t5*12i  interest  ? 

14.  If  50  men.  build  a  wall  750  rods  long  in  8  days, 
how  many  men  will  be  required  to  build  8 64 "5  rods  in  half 
of  the  time  ? 

15.  If  a  railroad  car  go  23  miles  in  45  minutes,  how  far 
will  it  go  in  5  days  of  10  hours  each  ? 

16.  If  in  247^  feet  there  are  15  rods,  how  many  rods 
in  1  mile  ? 

17.  If  47  acres  of  land  sell  for  $684'48,  what  will  be 
the  cost  of  a  farm  containing  287 '5  acres  ? 

18.  What  will  be  the  cost  of  847*56  pounds  of  wool,  if 
84-5  pounds  cost  $47-87|  ? 

19.  If  19  sheep  yield  56i  pounds  of  wool,  how  many 
pounds  will  387  sheep  yield  ? 

20.  How  many  pounds  of  coffee  can  be  bought  for 
$147-84,  when  18  pounds  cost  $l-93f  ? 

21.  If  a  tree  25  feet  4  inches  in  height  give  a  shadow 
of  50  feet  8  inches,  what  is  the  length  of  the  shadow  of 
a  tree  whose  height  is  84  feet  9  inches  ? 

Remark.— After  stating  the  question,  reduce  the  first  and  second  terms  to  the 
id^me  denominate  value. ;  also  reduce  the  J!/it>rf  term  to  its  lowest  denomination 
nien'ioned  : — the  answer  will  be  of  the  same  denomination. 


OPERATION. 

ft. 
25 

in.         ft. 

4  :     84 

in. 
9 

ft.    in. 

:  :  50  8  :     length 

of  shadow 

required. 

in. 
304 

in 

:  1017:: 
608 

in. 
608 

length  of  shadow  : 

required. 

8136 
6102 

304)618336(2034  in.  =  169  ft.  6  in.  Ana. 
608 


ART.  174.]  SIMPLE    PROPORTIO?i.  159 

22.  If  8  horses  eat  19  bushels  3  pks.  of  oats  in  a  week, 
how  much  would  85  horses  eat  iu  the  same  time  ? 

23.  If  12  men  in  6  weeks  earn  £145  IO5.  9d.,  how 
much  can  84  men  earn  in  half  of  the  time  ? 

24.  If  14  bushels  2  pks.  4  qts.  of  clover  seed  are  worth 
$6tl2i  how  much  will  184  bush.  3  pks.  6  qts.  cost? 

25.  If  15  horses  in  4  days,  consume  87  bush.  6  qts.  of 
■oats,  how  many  horses  will  610  bush.  1  pk.  2  qts.  keep  the 
same  time  ? 

26.  If  the  transportation  of  21  cwt.  147  miles,  cost 
$23*87^,  what  will  the  transportation  of  47  cwt.  3  qrs. 
20  lbs.  cost,  4  times  as  far  ? 

27.  If  a  person  accomplish  a  certain  piece  of  work  in 
242  days,  by  working  8  hrs.  a  day,  in  how  many  days  will 
he  accomplish  the  same  work,  by  working  12f  hours  a  day  ? 

28.  Allowing  a  person  to  perform  a  certain  journey  in 
26  days,  when  the  days  are  10^  hours  long;  in  what  time 
ought  he  to  accomplish  the  same  journey,  when  the  days 
are  13  hours  long  ? 

29.  Allowing  I3  A.  25  P.  of  land  to  produce  384  bush. 
3  pks.  of  wheat,  what  number  of  bushels  would  be  raised 
from  a  field  containing  47  A.  3  R.  30  P.,  at  the  same  rate  ? 

30.  An  army  of  4800  men  had  provisions  for  8  months, 
one-sixth  of  the  men  having  been  killed  in  battle,  how 
long  ought  the  same  provisions  last  the  remainder  ? 

31.  If  18  head  of  cattle  require  25  A.  3  R.  of  pasture 
ground,  during  the  summer,  how  many  acres  ought  36 
head  to  have  for  the  same  length  of  time  ? 

32.  Allowing  the  transportation  of  25  T.  18  cwt.  20  lbs., 
a  given  distance,  to  cost  $37*85;  how  much  should  be 
charged  for  the  transportation  of  18  T.  16  cwt.  3  qrs.^ 
10  lbs.  the  same  distance  ? 

33.  If  a  ship  sail  247  leagues  1  mile  6  fur.  in  15  days; 
in  how  many  days  would  she  sail  3000  miles  ? 

34.  A  borrowed  $250,  which  he  kept,  3  years  6  months. 
A  subsequently,  lends  B  $187|^.  How  long  ought  B  to 
keep  this  latter  sum,  in  return  for  the  accommodation  he 
afforded  A  ? 

35.  A  merchant  bought  3  pieces  of  cloth,  each  contain- 


160  PROPORTION.  [chap.  VII. 

ing  23  yds.  3  qrs.  for  $495"  15;  and  sold  54  yds.  2  qrs.  of 
it  for  what  it  cost.     How  much  did  he  receive  for  it  ? 

36.  If  8  yards  3  qrs.  of  cloth  cost  $34-50,  how  much 
will  83  E.  English  3  qrs.  cost  ? 

37.  If  5  E.  French  4  qrs.  of  cloth  cost  $14-60,  how  much 
will  12  yds.  3  qrs.  2  nas.  cost  ? 

38.  Allowing  14  horses  to  consume  65  bush.  3  pks.  5  qts. 
of  oats  in  a  week,  how  much  would  74  horses  consume  in 
the  same  time  ? 

39.  If  a  person  perform  a  certain  journey  in  14  days, 
by  traveling  9^  hours  a  day,  how  long  will  it  take  him  to 
perform  the  same  journey  by  traveling  12^  hours  a  day  ? 

40.  What  will  be  the  cost  of  9  cwt.  3  qrs.  20  lbs.  of 
beef,  if  8  cwt.  cost  $68  ? 

41.  If  36  sacks,  each  measuring  5  bushels,  contain  a 
eiven  quantity  of  grain ;  how  many  sacks,  each  containing 
3i  bushels,  will  contain  the  same  quantity  ? 

42.  Allowing  32  head  of  cattle  to  require  23  A.  3  R 
25  P.  of  pasture  ground,  during  the  summer,  how  many 
acres  will  145  cattle  require  for  the  same  length  of  time  ? 

43.  If  4  men  mow  7*97^  A.  of  grass  in  a  day,  how  many 
men  will  be  required  to  mow  63-8  A.  in  half  the  time  ? 

44.  The  capacity  of  a  cistern  is  3600  gallons,  and  is 
filled  with  water  by  a  pipe  which  pours  into  it  10  gals. 
3  qts.  a  minute.  By  a  leakage,  1  gal.  2  qts.  1  pt,  leaks 
out  every  minute  during  the  time  of  filling.  In  what  time 
will  the  cistern  be  filled  ? 

45.  If  f  of  an  acre  of  land  is  worth  $136,  how  much 
is  If  of  an  acre  worth  ? 

Rf.mark. — Questions  containing  fractions,  can  be  most  conveniently  salved 
b}'  finding  the  ratio  of  the  first  to  the  second  term,  and  then  multiply  the 
third  term  by  it.      (Art.  172,    Proposition  4.) 

OPERATION    BY    CANCELLATION. 

I  :  If  ::  $136  :  value  sought. 
3   68 

|X^X— =  $204  Ans. 


ART.    1T4.]  SIMPLE   PROPORTION.  161 

46.  If  f  of  a  farm  is  worth  $860,  how  much  is  f  of  it 
worth  ? 

47.  If  14  of  a  city  lot  is  worth  $4800,  how  much  is  "| 
of  it  worth  ? 

48.  If  I  of  a  barrel  of  flour  is  worth  $5-40,  how  much 
is  yV  ^^  i^  worth  ? 

49.  What  cost  16f  pounds  of  tea,  if  6f  pounds  cost) 
$8-55  ? 

50.  What  length  of  board  that  is  16y^3  inches  in  width, 
will  be  required  to  make  a  square  foot  ? 

51.  Bought  15^  yards  of  cloth  for  $54*90  ;  what  will 
25  yards  3  qrs.  cost  at  the  same  rate  ? 

52.  If  f  of  a  ship  is  worth  $34865,  how  much  is  the 
whole  cargo  worth  ? 

53.  If  j\  enough  water  run  into  a  ship  by  a  leak,  in  1 
day  9  hrs.  15  miu.,  to  sink  her ;  how  long  before  she  will 
sink  ? 

54.  Bought  25|  barrels  of  flour,  at  $6y«y  a  barrel,  and 
paid  for  it  with  sheep,  at  $li  a  head  ;  how  many  sheep 
did  it  take  ? 

55.  If  6|  barrels  of  sugar  cost  $112.15,  how  much  will 
A  of  a  barrel  cost  ? 

56.  If  13f  yards  of  cassimere  cost  $19|,  what  will  5| 
yards  cost  ? 

51.  If  2f  barrels  of  beef  cost  $20-*I5,  how  much  will  1^ 
barrels  cost  ? 

58.  If  5  pounds  of  butter  cost  62i  cents,  how  much 
will  If  pounds  cost. 

59.  If  I  of  an  apple  cost  |  of  a  cent,  what  will  |  of  an 
apple  cost  ? 

60.  If  it  require  6  days  for  10  men  to  build  360  rods  of 
wall,  how  many  men  can  in  i  of  the  time  build  120  rods 
of  similar  wall  ? 

61.  If  24  men  in  8  days  perform  a  certain  piece  of  work, 
how  many  men  will  be  necessary  to  accomplish  3  times  as 
much  work  in  |  of  a  day  ? 

62.  If  it  require  2  bushels  of  oats  to  feed  4  horses  ^  of  a 
day,  how  many  horses  would  it  take  to  consume  144  bushels 
in  f  of  a  day  ? 


162  PROPORTION.  [chap.    VII 

63.  If  a  staff  9f  feet  long  cast  a  shadow  12|  feet,  what 
is  the  height  of  that  steeple  the  shadow  of  which,  at  the 
same  time  measures  285  feet  ? 

64.  If  a  steamship  can  sail  3000  miles  in  9^  days,  how 
long,  at  the  same  rate  of  sailing,  would  she  require  to 
sail  24900  miles,  the  distance  around  the  earth  ? 

65.  Tiie  diurnal  rotation  of  the  earth  moves  its  equato- 
rial portions  about  24900  miles  a  day.  (24  hours.)  How 
far  is  that  in  each  minute  ? 

66.  Admitting  the  earth  to  move  in  its  orbit  about  the 
sun  59*1000000  miles,  in  365  days  6  hours  ;  how  far  on  an 
average  does  it  move  in  1  minute  ? 

67.  If  it  require  35  yards  of  carpeting,  which  is  |  of  a 
yard  wide  to  cover  a  floor,  how  many  yards,  which  is  1^ 
yards  wide,  will  be  necessary  to  cover  the  same  floor  ? 


COMPOUND  PROPORTION. 

Art.  ITo.  Compound  Proportion  teaches  to  find  a 
required  quantity  in  a  proportion  when  it  depends  on  more 
than  three  terms. 

1.  If  6  men  can  earn  $72  in  10  days,  by  working  12 
hours  a  day,  how  many  dollars  can  15  men  earn  in  8  days, 
by  working  8  hours  a  day  ? 

Remark. — We  will  first  solve  this  question  by  analysis. 

Analysis If  6  men  in  a  certain  time  earn  $72,  1  man  in 

the  same  time  will  earn  ^  of  $72  =  $12 ;  and  15  men  will  earn 
15  times  $12  =  $180.  If  in  10  days  15  men  earn  $180.  in  1  day 
they  wiU  earn  yV  of  $180  =  $18  ;  and  in  8  days  they  will  earn 
8  times  $18  =  $144.  If  in  8  days  by  w^orking  12  hours  a  day, 
15  men  earn  $144,  by  working  1  hour  a  day,  they  will  earn 
yV  of  $144  =  $12 ;  and  by  working  8  hours  a  day  they  will 
earn  8  times  $12  =  $96. 

SOLUTION    BY    CANCELLATION. 

IIkmakk  — As  the  question  is  read  the  pupil  will  find  it  of  assistance  to 
write  it  down  in  the  following  manner,  as  he  can  then  more  easily  remember 
the  question  and  form  the  ratios.  Taking  the  above  example  we  proceed 
thus  ;— 


kRT.    115.]  '  COMPOUND    PROPOKTION.  163 

men.         $.         days,      hours. 

6       12       10       12 
16 8         8 

X^         3         4 

Explanation If  6  men  in  a  certain  time  earn  $72,  1  man 

will  earn  \  of  $72^  and  15  men  will  earn  y  of  $72.  If  15  men 
earn  y  of  $72  in  10  days,  in  1  day  they  will  earn  jL  as  much, 
and  in  8  days  y^^  as  much,  which  is  ^-^  of  ^  of  $72.  If  15  men 
in  8  days  earn  y\  of  */  of  $72  by  working  12  hours  a  day,  by 
working  1  hour  a  day  they  will  earn  y'^  as  much,  and  by  working 
8  hours  a  day  y^2  as  much,  which  is  y\  of  y^^  of  ^-^  of  $72  =  $96. 

2.  If  12  men  can  mow  48  acres  of  grass  in  8  days,  by 
working  5  hours  a  day;  how  many  acres  can  56  men  mow 
in  5  days,  by  working  12  hours  a  day  ? 

3.  If  the  wages  of  36  men  for  3  days  be  $216;  how 
many  men  in  4  days  can  earn  $192  ? 

4.  If  15  men  can  cut  280"  cords  of  wood  in  16  days,  by 
working  9  hours  a  day,  how  many  men  will  be  required  to 
cut  28  cOtds  in  4  days,  by  working  6  hours  a  day  ? 

5.  If  a  man  travel  240  miles  in  14  days,  by  traveling 
6  hours  a  day;  how  far  can  he  travel  in  18  days,  by 
traveling  9^  hours  a  day  ? 

6.  If  15  men  in  9  days,  by  working  6  hours  a  day,  build 
36  rods  of  stone-fence;  how  many  men  will  be  required  to 
build  133  3  rods  in  14  days,  by  working  8  hours  a  day  ? 

7.  If  72  men  in  18  days  of  12  hours  each,  build  a  wall 
162  rods  in  length,  12  feet  high,  and  9  feet  thick;  how 
many  rods  of  wall  that  is  9  feet  high,  and  3  feet  thick,  can 
40  men  build  in  8  days  of  9  hours  each  ? 

8.  If  a  marble  slab  20  feet  long,  6  feet  wide,  and  4  inches 
thick,  weigh  850  pounds;  what  is  the  length  of  another 
slab  that  is  4  feet  wide  and  2  inches  thick,  that  weighs 
212  pounds  ? 

9.  If  a  family  of  12  persons  in  20  weeks  and  4  days  con- 
sume $450  worth  of  provisions;  how  many  persons  will 
$803 7 1  worth  of  provisions  keep  45  weeks  and  6  days  ? 


164  PROPORTIOX.  [chap.   VII. 

10.  If  it  require  264  yds.  of  cloth  tliat  is  li  yds.  wide, 
to  clothe  121  men;  how  many  yards  which  is  li  yards 
wide  will  be  required  to  clothe  220  ? 

11.  If  210  yds.  of  cloth,  1  yard  wide,  cost  $300,  what 
will  140  yds.  of  similar  cloth  cost,  that  is  3  quarters  wide  ? 

12.  If  $250  will  in  7  months  gain  $25,  when  the  rate  of 
interest  is  10  per  cent.;  at  what  rate  per  cent.,  will  $750 
in  9  months  ^ain  $67^  ? 

13.  If  a  family  of  24  persons  consume  $120  worth  of 
bread  in  8f  months,  when  flour  is  worth  $5  a  barrel;  how 
many  dollar's  worth  will  a  family  of  8  persons  consume  in 
6  months,  when  flour  is  worth  $7  a  barrel  ? 

14.  If  240  men,  by  working  8  hours  a  day,  can  in  81 
days  dig  256  cellars,  each  24  feet  long,  27  feet  wide,  and 
18  feet  deep;  how  many  men  can,  in  27  days  of  6  hours 
each,  dig  18  cellars,  each  40  feet  long,  36  feet  wide,  and 
12  feet  deep  ? 

15.  If  24  men,  by  working  8  hours  a  day,  can  in  18  days 
dig  a  ditch  95  rods  long,  12  feet  wide,  and  9  feet  deep, 
how  many  men,  by  working  12  hours  a  day,  for  24  days, 
will  be  required  to  dig  a  ditch  380  rods  long,  9  feet  wide 
and  6  feet  deep,  in  a  soil  that  is  1|  times  as  difficult  of 
excavation  ? 


CONJOINED  PROPORTION. 

Art.  176.  Conjoined  Proportion  is  a  proportion  in  which 
each  antecedent  is  equal  in  value  to  its  consequent, — each 
consequent  being  of  the  same  denomination  as  the  preceding 
antecedent, — and  the  first  and  last  terms,  of  the  same  denom- 
ination. 

1.  If  8  bushels  of  wheat  are  worth  3  cords  of  wood,  and 
9  cords  of  wood  are  worth  3  tons  of  hay,  }\ow  many  bushels 
of  wheat  are  worth  6  tons  of  hay  ? 

Analysis. — If  3  tons  of  hay  are  worth  9  cords  of  wood,  1  ton 
is  worth  §  cords  of  wood.    If  3  cords  of  wood  are  worth  8  bush. 


ART.  lYt.]  COPARTNERSHIP,  165 

of  wheat,  1  cord  .*s  worth  |  bushels.  If  1  .^ord  is  worth  |  bush, 
of  wheat,  |  cords  (the  value  of  1  ton  of  hay),  is  worth  |  times 
I  bushels  of  wheat,  and  6  tons  are  worth  6  times  f  X  f  bushela 
=  48  bushels  of  wheat. 

The  conditions  of  the  above  question  are  expressed  thus : 

8  bushels  =  3  cords  of  wood. 

9  cords  =  3  tons. 

6  tons  =  how  many  bushels  of  wheat  ? 

And  may  be  solved  by  writing  all  the  terms  on  the  left  of  the 
equality,  for  the  numerator  of  a  compound  fraction,  and  those 
on  the  right  for  the  denominators.     Thus : 

$ 

6     0     8 

tXaXs  =  48  bushels  of  wheat. 

2.  If  4  barrels  of  corn  are  worth  8  bushels  of  wheat,  and 
3  bushels  of  wheat  are  worth  5  bushels  of  rye,  and  12 
bushels  of  rye  are  worth  20  bushels  of  oats,  how  many 
bushels  of  oats  are  worth  12  barrels  of  corn  ? 

3.  A  can  do  as  much  work  in  3  days  as  B  can  in  6  days ; 
and  B  as  much  in  5  days  as  C  in  15  days.  In  how  many 
days  could  A  do  as  much  work  as  C  in  48  days  ? 

4.  If  48  yards  of  cloth  in  New  York  are  worth  36  bar- 
rels of  flour  in  Philadelphia ;  and  18  barrels  of  flour  in 
Philadelphia  are  worth  24  bales  of  cotton  in  New  Orleans; 
how  many  bales  of  cotton  in  New  Orleans  are  worth  240 
yards  of  cloth  in  New  York  ? 

5.  If  121  yards  of  satin  cost  $18*I5;  and  $10-25  will 
purchase  3  yards  of  broadcloth;  and  6^  yards  of  broadcloth 
are  worth  18^  yards  of  silk;  how  many  yards  of  satin  are 
worth  120  yards  of  silk  ? 


COPARTNERSHIP.* 

Art.  177.  Copartnership  is  the  association  of  two  or 
more  individuals  in  the  transaction  of  business,  who  agree 

*  CopartntrshtPi  or  Fellowship,  is  sometimes  called   PAKimvi;  Proportiow. 


166  PROPORTION.  [chap.    VII. 

to  share  the  profits  and  losses  in  proportion  to  the  amount 
of  capital  they  have  in  the  partnership.  Each  individual 
thus  associated  is  called  a  Partn&r.  The  partners  together 
are  called  the  Comjpany,  or  Firm. 

The  Capital  Stock  is  the  amount  of  money  employed  in 
trade.     The  Dividend  is  the  profit  or  loss  to  be  shared. 

1.  A,  B,  and  C  entered  into  partnership.  A  put  in 
$240;  B  put  in  $400;  and  C  put  in  $320.  They  gain  $192. 
How  much  is  each  man's  gain  ? 

OPERATION, 

A's  stock,  $240 
B's  "  400 
C's      "        320 

Capital  stock,    960 
Therefore,  A  owns  f  f  ^  =  :|-  of  the  entire  stock. 

BU          40  0    _5_  u  u 

Q^TT  12 

C4;         320    1  U  (( 

9(J0   —   3 

Hence,  As  gain  is  \  of  $192  =  $48 
B's  "  j%  of  $192  =  $80 
C's       "       i  of  $192  =  $64 

2.  A,  B,  and  C  enter  into  partnership.  A  puts  in  $360; 
B  puts  in  $440;  and  C  puts  in  $500.  They  gain  $t80. 
How  much  is  each  man's  gain  ? 

8.  A,  B,  C,  and  D,  hired  a  pasture  for  $12:  A  put  in 
12  sheep;  B  put  in  16;  C  18;  and  D  14.  How  much 
ought  each  to  pay  ? 

4.  Four  men  traded  in  company  and  gained  $1680  . 
A's  stock  was  $2000;  B's  $1600;  C's  $2400;  and  D's 
$2000.     How  much  is  each  man's  gain  ? 

5.  A  farm  was  purchased  for  $7000,  by  A,  B,  and  C. 
A  furnished  $2500;  B  $3000;  and  C  $1500.  They  re- 
ceive $560  rent  yearly.  How  much  of  this  rent  should 
each  receive  ? 

6.  A  merchant  employed  4  clerks,  at  the  annual  salaries 
of  $250,  $300,  $400,  $500,  respectively.  At  the  end  of 
the  year  the  merchant  proving  bankrupt,  has  but  $870  to 


ART.    17 1.]  COPARTNERHIP.  16t 

be  divided  proportionally  among  them.     What  will  be  the 
portion  of  each  ? 

7.  Divide  $960  among  three  persons  in  such  a  manner 
that  their  shares  shall  be  to  each  other  as  5,  4,  and  3  re- 
spectively ? 

8.  Two  persons  form  a  partnership  in  trade,  with  a  cap- 
ital of  $1500,  of  which  the  first  contributed  $940;  and 
the  second  the  remainder.  They  gain  $640.  How  much 
is  each  one's  share  ? 

9.  Divide  the  number  230  into  three  parts  which  shall 
be  to  one  another  as  i,  |,  and  f . 

Analysis. — the  proportional  terms  being  reduced  to  equiva- 
lent fractions  having  a  common  denominator,  we  have  ~,  y^g, 
and  y^2  ;  ^'^d  these  fractions  are  to  one  another  as  their  nume- 
rators 6,  8,  and  9^  since  they  have  the  same  denominator. 
Hence  we  divide  the  230  into  6  -}-  8  -|-  9  =  23  equal  parts. 
Hence  2%,  aV  ^^^  2^3  ^^  ^^^  respectively,  gives  the  required 
numbers. 

10.  A,  B,  and  C,  found  a  purse  containing  $240,  and 
agreed  to  share  it  in  the  proportion  of  |,  ^,  and  f .  How 
much  should  each  receive  ? 

11.  A,  B,  and  C  enter  into  partnership:  A  puts  in 
$160;  B  $280;  and  C  $460.  They  lose  $480.  How 
much  is  each  partner's  loss  ? 

12.  A  captain,  mate,  and  14  sailors,  took  a  prize  of 
$24600;  of  which  the  captain  takes  11  shares;  the  mate 
5  shares  ;  and  the  remainder  is  equally  divided  among  the 
sailors.     How  much  did  each  receive  ? 

13.  Four  partners,  A,  B,  C,  and  D  shipped  1280  sheep 
for  Scotland;  of  which  A  owned  240;  B  160;  C  400; 
and  D  the  remainder.  In  a  severe  storm  they  threw  320 
of  them  overboard.  How  many  sheep  did  D  own,  and 
how  much  was  each  partner's  loss  ? 

14.  A,  B,  C,  D,  and  E  are  to  share  $3045;  A  is  to 
have  a  certain  sum;  B  as  much  again  as  A;  C  as  much 
as  A  and  B  together;  D  as  much  again  as  B;  and  E  as 
much  as  D  and  A  together.     How  much  is  each  to  have  ? 

15.  A,  B,  and  C  agree  to  contribute  $620'62  towards 
building  a  church,  which  is  to  be  situated  2  miles  from  Aj 


168  PROPORTION.  [chap.    VII. 

3  miles  from  B;  and  5  miles  from  C.  They  also  agree 
that  their  contributions  shall  be  proportional  to  the  recip- 
rocals of  their  distances  from  the  church.  How  much 
9ught  each  to  contribute  ? 

16.  A,  B,  and  C  contribute  $3535- tO  towards  building 
an  Academy,  which  is  to  be  situated  1^  miles  from  A; 
If  miles  from  B;  and  21  miles  from  C.  They  also  agree 
that  their  contributions  shall  be  reciprocally  proportional 
to  their  distances  from  the  Academy.  How  much  did 
each  contribute  ? 

It.  A,  B,  and  C  found  a  purse  containing  $280' tO. 
They  agreed  to  divide  it  in  such  a  manner  that  A  should 
rave  I  as  much  as  B;  and  B  |-  as  much  as  C.  How  much 
should  A,  B,  and  C  receive  respectively  ? 

Rkmark. — The  pupil  will  find  the  proportional  terms  as  follows  : 

A's  part  =  I  of  B's, 
and  B's  =  |  of  C's 

Hence,  |  of  B's  =    C. 

Therefore,  A's  =  y\  of  B's 
B's  =  if 

C's  =  If  of  B's.     Consequently  we  divide 
the  $280*70  in  proportion  to  the  numbers  8,  12,  and  15. 

18.  A,  B,  and  C,  in  partnership  lose  $650.  A's  por- 
tion of  the  capital  employed  was  |  of  B's,  and  B's  was  f 
of  C's.     What  amount  of  loss  should  each  sustain  ? 

19.  Four  persons  in  a  joint  speculation  gain  $460,  which 
is  to  be  divided  among  them  so  that  the  second  shall  have 
2-  as  much  as-  the  first,  and  the  second  f  as  much  as  the 
third.     How  much  should  each  receive  ? 

20.  A  farmer  divided  1152  acres  of  land  among  his  four 
sons,  in  such  a  manner  that  |  of  John's  number  of  acres 
equals  f  of  James';  |  of  Jame's  equals  f  of  Jackson's; 
and  I  of  Jackson's  equals  f  Joseph's  number  of  acres. 
How  many  acres  did  each  receive  ? 

21.  Three  men  A,  B,  and  C,  agree  to  reap  a  certain 
field  of  wheat,  for  $39-68  ;  A  and  B  calculate  that  they 
can  do  f  of  the  labor  ;  A  and  C,  that  they  can  do  |  ;  and 
B  and  C  that  they  can  do  |  of  it.  How  much  can  each 
receive  according  to  these  estimates  ? 


ART.    118.]  COMPOUND    COPARTNERSHIP.  169 


COMPOUND  COPARTNERSHIP. 

Art.  178.  When  the  stock  of  the  several  partners  is  em- 
ployed in  the  trade  for  different  periods  of  time,  it  is  called 
Compound  Copartnership.  It  is  evident  in  such  cases,  that 
the  gai7i  or  loss  must  be  apportioned  with  reference  to  the 
stock  and  the  time  it  has  been  employed  in  the  business. 

1.  Three  partners  A,  B,  and  C  put  money  into  trade  as 
follows  :  A  put  in  $50  for  4  months  ;  B,  $150  for  2 
months  ;  and  C,  $250  for  3  months.  They  gained  $250. 
How  much  is  each  man's  share  of  the  gain  ? 

OPERATION. 

$  m.  $ 

50  X  4  =  200  for  1  month. 
150  X  2  =  300   " 
250  X  3  =  750   " 


1250  Capital  Stock. 
Explanation. — The  preceding  work  becomes   evident,   by 
considering  that  the  interest  of  $50  for  4  months,  is  the  same 
as  the  interest  of  $200  for  one  month ;  &c.     Therefore, 
A's  part  of  the  entire  stock  =  -f^^^  =  ^g  of  the  whole. 

^°  T2-5O   3 

Hence,  A's  gain  =  /^  of  $250  =  $40 
B's  "  =  2^  of  $250  =  $60 
C's    "     =  I  of  $250  =    $150. 

2.  A,  B  and  C  hire  a  pasture  for  $240  ;  A  put  in  16 
cows  for  10  weeks  ;  B,  20  cows  for  1  weeks  ;  and  C,  25 
cows  for  6  weeks.     How  much  ought  each  to  pay  ? 

3.  A,  B,  C,  and  D  have  together  performed  a  piece  of 
work,  for  which  they  receive  $266*40.  A  worked  16  days 
of  10  hours  each  ;  B  worked  20  days  of  12  hours  each; 
C  worked  14  days  of  1 0  hours  each;  and  D  worked  15  days 
of  12  hours  each.     How  much  should  each  man  receive  ? 

4.  A,  B,  C,  and  D  engaged  in  partnership  for  3  years. 
A  advanced  $2500,  B  $3500,  C  and  D,  each  $3800. 
Nine  months  afterwards,  A  added  $600  to  his  stock  ;  B 

8 


ItO  PROPORTION-.  [chap.    VII 

$350;  C  withdrew  $180  ;  anclD  withdrew  |460.  At  the 
end  of  the  3  years,  the  profits  were  found  to  be  $1200 
How  much  is  each  one's  share  ? 

^5.  To.  gather  a  certain  field  of  grain,  A  furnished  9 
laborers  6  days  ;  B  12  laborers  for  4  days  ;  and  C  14 
laborers  for  5  days.  For  the  whole  work  they  received 
$54*85.  How  much  should  A,  B,  and  C  receive  respec- 
tively ? 

6.  An  army,  consisting  of  3  generals,  5  colonels,  12 
captains,  and  6840  soldiers,  took  a  prize  of  $89908*15, 
which  they  agree  to  divide  among  themselves  in  propor- 
tion to  their  pay  and  the  time  they  have  been  in  the 
army.  The  generals  and  colonels  have  been  in  the  army 
9  months  ;  the  captains  5  months ;  and  the  soldiers,  8 
months  ;  the  generals  have  $60  a  month  ;  the  colonels, 
$40  ;  the  captains,  $15  ;  and  the  soldiers,  $10.  How 
much  ought  each  to  receive  ? 

ALMGATION  MEDIAL.* 

Art.  179.  Alligation  Medial  teaches  the  method  of 
finding  the  average  value  of  a  mixture  when  the  several 
simples  of  which  it  is  composed,  and  their  values  are 
known. 

Art.  180,  Given  the  several  ingredients  and  theii 
respective  values  to  find  the  average  value  of  the  compound 

1.  A  farmer  mixes  together  10  bushels  of  oats,  worth 
40  cents  a  bushel;  15  bushels  of  corn,  worth  50  cents  a 
bushel;  and  25  bushels  of  rye,  worth  70  cents  a  bushel. 
What  is  the  value  of  a  bushel  of  the  mixture  ? 


OPERATION. 

cts. 

bush. 

cts. 

40 

X  10  = 

400 

50 

X  15  = 

750 

70 

X  25  = 

1750 

50  ) 

2900 
58  cts. 

*  Alligation  Medial  is  sometimes  called  Medial  Proportitn. 


ART.    182.]  ALLIGATION   MEDIAL.  .  xTl 

cts.     hush.         ct..  EXPLANATION. 

40  X  10  =   400  )•  10  bush,  at  40  cts.  a  bushel  is  worth    $4-00. 

50  X  15  =    750  }  15  bush,  at  50  cts.  a  bushel  is  worth    $7'50. 

70  X  25  =  1750  }■  25  bush,  at  70  cts.  a  bushel  is  worth  $17*50. 

50  )  2900  }  $29  is  the  entire  cost  of  the  mixture^  which 

58  cts      l>6i°g  divided  by  50,  the  whole  number 

of  bushels,  gives  58  cents,  the  average 

value  of  1  bushel. 

2.  A  wine  merchant  mixed  together  40  gallons  of  wine, 
at  80  cents  a  gallon;  25  gallons  of  brandy,  at  ^0  cents  a 
gallon;  and  15  gallons  of  wine,  at  $1*50  a  gallon.  What 
is  the  value  of  a  gallon  of  the  mixture  ? 

3.  A  grocer  mixed  80  gallons  of  rum,  worth  30  cents  a 
gallon;  40  gallons  of  whiskey,  worth  40  cents  a  gallon; 
and  20  gallons  of  water,  at  the  usual  price.  What  is  the 
value  of  a  gallon  of  the  mixture  ? 

•  4.  A  grocer  mixed  120  pounds  of  sugar,  worth  5  cents 
a  pound;  150  pounds,  worth  6  cents  a  pound;  and  130 
pounds,  worth  10  cents  a  pound.  What  was  the  average 
value  of  a  pound  of  the  mixture  ? 

5.  A  grocer  sold  50  barrels  of  flour,  at  $7'20  a  barrel ;  70 
barrels,  at  $8*20  a  barrel;  and  80  barrels,  at  $5-70  a  barrel. 
How  much  on  an  average  did  he  receive  for  a  barrel  ? 

ALLIGATION  ALTERNATE. 

Art.  181.  Alligation  Alternate  teaches  the  method 
of  finding  how  much  of  several  ingredients,  the  values  of 
which  are  known,  must  be  taken  to  make  a  compound  of 
a  certain  value. 

CAsr  I. 

Art.  182.  Given  the  values  of  several  ingredients,  to 
make  a  compound  of  a  given  value.  First,  Place  the  several 
values  of  the  ingredients  in  a  column,  and  the  average  value 
on  the  left  of  this  column.  Join  with  a  curved  line,  the  value 
of  each  ingredient  that  is  less  than  the  average  value,  with  one 
or  more  that  is  greater  ;  then  place  the  difference  between  the 
value  of  each  ingredient  and  the  average  value,  opposite  the 
price  of  the  ingredient  with  which  it  is  joined,  and  this  dif 


1*72  PROPORTION.  [chap.    VII. 

ference,  or  the  sum  of  these  differences,  (if  there  is  more  than 
one,)  will  he  thz  quantity  required  of  that  ingredient. 

1.  How  much  sugar  worth  6,  8,  and  10  cents  a  pound, 
must  be  mixed  together,  so  that  a  pound  of  the  mixture 
may  be  worth  7  cents  ? 


(10^     = 


OPERATION. 

=  1  -^  3  =  4  of  the  sugar,  at  6  cts.  a  pound. 
1     "          "        "     8  cts.      " 

1     "  "        "  10  cts.       " 


Explanation.— By  taking  one  pound  of  each  kind  of  the 
sugar,  we  shall  receive  on  the  10  cent  quality,  4  cents  more 
than  the  average  price  of  the  mixture,  and  on  the  6  cent  qual- 
ity 1  cent  less  than  the  average  price.  The  gain  and  the  loss 
on  the  different  qualities  of  sugar  are  to  be  equal ;  therefore 
the  quantities  taken  must  be  universally  proportional  to  the 
gain  and  the  loss  on  the  respective  qualities. 

Remark. — Questions  of  this  kind  admit  of  an  iruhfinitt  number  of  answers. " 
It  is  obvious  if  we  take  any  other  quantities  which  are  to  each  other,  as  4,  1 
and  1  :  as  8,  '2  and  2  ;  12,  .3  and  3.  &c.,  that  they  will  each  satisfy  the  condi- 
tion of  the  question  equally  well. 

It  is  evident  that  there  may  be  as  many  answers  of  diflerent  ratios,  as  there 
are  methods  of  connecting  the  several  values  of  the  ingredients.    For  example: 

2.  How  many  pounds  of  tea,  at  5,  6,  9,  and  12  shillings 
4  pound,  must  be  mixed,  so  that  the  mixture  shall  be 
worth  8  shillings  a  pound  ? 


ART.    183.]  ALLIGATION    ALTERNATE.  ITS 

3.  How  much  wine,  at  $110  per  gallon,  60  cents  per 
gallon,  and  40  cents  per  gallon,  must  be  mixed  together, 
so  that  the  mixture  may  be  worth  80  cents  per  gallon  ? 

4.  How  much  wine,  at  $r60  a  gallon,  and  water,  at  the 
usual  rate,  must  be  jnixed  together,  so  that  the  compound 
may  be  worth  $ri5  a  gallon. 

5.  How  much  of  each  sort  of  grain,  at  46,  54,  t5,  and 
85  cents  a  bushel,  must  be  mixed  together  so  that  the 
compound  may  be  worth  65  cents  a  pound  ? 

CASE    II. 

Art.  183.  When  one  of  the  ingredients  is  limited  to  a 
giyen  quantity. 

1.  A  merchant  wishes  to  mix  60  pounds  of  tea,  worth 
$1"20,  with  three  other  kinds,  worth  $110,  70  cents,  and 
60  cents,  a  pound,  respectively,  so  that  the  mixture  may 
be  worth  $0-80  a  pound.  How  many  pounds  of  the  last 
three  kinds  must  be  used  ? 

OPERATION. 

go  I  110x^  =  10'       3  _J    30'    "  "      SMO 


r  120-^  =20  ^  r    60  pounds,  worth  $120  a  pound. 

J  llOx   \  =  10  '  ^  o  _1    30       "  "      SMO 

1    70>'y  =  30  f  ^  '^  — 1    90       "  "      $0-70 

[    60-^  =40  J  [120       "  ''      $0-60        •' 

Explanation. — By  Case  1,  we  obtain  20,  10,  30,  and  40 
pounds,  respectively,  which  meets  the  requirements  of  the 
question,  were  neither  of  the  quantities  limited  ;  but  there  is  to 
be  60  pounds  of  that  which  is  worth  $1*20  a  pound.  We  there- 
fore, multiply  the  20  opposite  the  Sl-20  by  such  a  number  as 
will  cause  the  product  to  become  60,  which  I  find  to  be  3,  and 
io  preserve  the  value  of  the  mixture  the  same  per  pound,  we 
multiply  all  the  other  proportional  quantities  by  the  same 
number. 

2.  How  much  oats,  at  $*40  a  bushel ;  barley,  at  $'45; 
and  corn,  at  $'75,  must  be  mixed  with  60  bushels  of  rye,  at 
$'85  a  bushel,  so  that  a  bushel  of  the  mixture  may  be  worth 
$•60? 

3.  How  much  sugar,  at  5,  8,  and  10  cents  a  pound  must 
be  mixed  with  64  pounds,  at  12  cents  a  pound,  so  that  the 
mixture  may  be  worth  9  cents  a  pound  ? 


114  PBOPORTION.  [chap     VII 

4.  A  merchant  has  40  pounds  of  tea,  worth  $150  a 
pound,  which  he  wishes  to  mix  with  four  other  kinds,  worth 
95,  *I5,  60,  and  40  ctjuts  a  pound  respectively.  Ilow  much 
must  he  take  of  each  of  these  four  kinds,  so  that  the  mix- 
ture shall  be  worth  80  cents  a  pound,? 

CASE    III. 

Art.  184.  When  the  whole  mixture  is  to  consist  of  a 
certain  quantity. 

1.  A  merchant  has  sugar  worth  5,  6,  9,  and  12  cents  a 
pound; — with  a  mixture  of  these  he  wishes  to  fill  a  hogs- 
head that  shall  contain  220  pounds.  How  much  of  each 
kind  must  he  take,  so,  that  the  compound  may  be  worth 
8  cents  a  pound  ? 

OPERATION. 

r    5-^  =4  ]  (  88  pounds  at  5  cents  a  pound.  1 

8  t^zll  22=  ^1   ::    t  ::     ::  Un. 

il2-^=3J  166        '^       12     «  "      J 

"10)220 

22     J  Ratio  of  the  sum  of  the  proportionate  quantities  to 
(      the  number  given. 

Explanation — The  sum  of  the  proportionate  quantities, 
(found  by  Case  1.)  is  10 ;  the  whole  number  of  pounds  that  is 
to  compose  the  mixture,  is  220 ;  therefore,  I  must  take  ^f^* 
times  as  much  as  the  sum  of  these  proportionals,  which  is  22 
times  each  proportionate  quantity. 

2.  How  many  gallons  of  water,  brandy,  and  rum,  must 
be  taken,  so  as  to  make  a  mixture  of  90  gallons,  worth  80 
cents  a  gallon;  providing  the  water  is  of  no  value,  the 
brandy  being  worth  |1'20  a  gallon,  and  the  rum,  60  cents 
a  gallon  ? 

Rkmark. — Archimedes  employed  the  above  in  detecting  the  fraud  respecting 
the  crown  of  Hiero,  king  of  Syracuse.  The  king  had  ordered  a  crown  of  i)ure 
gold  to  be  made  ;  but  s-ispecting  his  artist  to  have  mixed  alloy  with  it,  he  re- 
quested Archimedes  to  d«^termine  the  fact  without  injuring  the  crown.  To 
do  this,  Archimedes  tookr  a  piece  of  pure  gold,  and  another  of  alloy,  each 
equal  in  weight  to  the  crown,  placing  them  respectively  in  a  vessel  filled 
with  water  and  observing  the  quantity  of  water  expelled  by  each  he  readily 
determined  that  the  crown  was  composed  of  gold  and  alloy  ;  also  the  exact 
proportion  in  which  these  ingredients  were  utfed. 


ART,  18  T.J  PERCENTAGE.  175 

3.  Suppose  the  weight  of  the  crown  and  of  each  mass 
to  be  10  pounds  ;  and  that  being  placed  in  water,  the  alloy 
expelled  "92  lbs.,  the  gold  .52  lbs.,  and  the  crown  "64  lbs. 
Of  how  much  gold,  and  of  how  much  alloy,  did  the  crown 
consist  ?  Ans.  3  lbs.  of  alloy,  and  7  lbs.  of  gold. 

OPERATION. 

•40)10-00 
25 


CHAPTER  VIII. 
PERCENTAGE 


Art.  185.  The  term  per  cent,  is  derived  from  the 
Latin  words  per  and  centum,  which  signify,  by  ike  hundred. 
Percent,  therefore,  is  any  sum  or  number  on  a  hundred, 
whatever  be  the  denomination.     Thus,  5  per  cent.,  signifies 

5  for  every  hundred,  or  5  hundredths ;  8  per  cent,  signifies 
8  for  every  hundred,  or  8  hundredths,  &c. 

We  have  already  learned  that  hundredths  can  be  ex- 
expressed  either  as  a  wnmon  or  as  a  decinal  fraction  ;  thus, 

6  hundredths  =yf^  =  .05;  8  hundredths^  yf-  =  .08,  &c. 
In  all  our  calculations  in  percentage,  the  rate  per  cent,  is 
written  in  the  decimal  form. 

Art.  186.  Percentage  is  extensively  used  in  mercan- 
tile transactions,  and  more  or  less  in  the  transactions  of 
all  other  kinds  of  business  ;  such  as  Assessment  of  Taxes, 
Insurance,  Puties,  Profit  and  Loss,  Interest,  Discount, 
&c.,  &c. 

Art.  187.  Finding  the  percentage  on  any  sum  or 
quantity. 

1.  What  is  5  per  cent,  of  225  barrels  of  sugar  ? 


It6  PERCENTAGE.  [CHAP.    VIIl, 


OPERATION. 
225 

.05 


1125 

It  may  also  be  solved  thus ;  5  per  cent,  is  yl^  =  oV  of  the 
given  quantity.  Therefore,  3^  of  225  barrels  =  11-25  barrels, 
is  5  per  cent,  of  225  barrels. 

2.  What  is  6  per  cent,  of  $140  ? 

3.  What  is  8  per  cent,  of  $340  ? 

4.  What  is  35  per  cent,  of  $380  ? 

5.  What  is  47  per  cent,  of  $160'35  ? 

6.  What  is  12|  per  cent,  of  146  yards  of  cloth  ? 

I.  What  is  14|  per  cent,  of  864  gallons  of  molasses  ? 

8.  What  is  16|  per  cent,  of  8472  barrels  of  flour  ? 

9.  A  man,  having  $9684,  lost  by  an  investment  12| 
per  cent,  of  it ;  how  much  had  he  remaining  ? 

10.  Bought  24  head  of  cattle  at  $25  a  head,  and  sold 
them,  at  25  per  cent,  advance ;  how  much  did  I  gain  ? 

II.  A  merchant  having  $8645,  gave  14  per  cent,  of  it 
for  silks;  28  per  cent,  of  it  for  flour;  43  per  cent.^of  it 
for  broadcloth;  and  the  remainder  for  sugar.  How  many 
dollars  did  he  spend  for  each  ? 

12.  A  farmer  raising  98 T  bushels  of  wheat,  gives  9  per 
cent,  of  it  forgathering  it;  10  per  cent,  of  the  remainder 
for  thrashing;  and  10  per  cent,  of  what  now  remains  for 
flouring.     How  much  has  he  remaining  2 

13.  A  merchant  bought  563  barrels  of  cider  for  $2837; 
and  sold  45  per  cent,  of  it,  at  $6-85  a  barrel ;  35  per  cent. 
of  it,  at  $7'12i  a  barrel;  and  the  remainder  for  what  it 
cost.     How  much  did  he  gain  by  the  operation  ? 

14.  A  speculator  invested  $8640  in  a  speculation,  and 
lost  25  per  cent.;  he  then  invested  the  remainder  in  a 
speculation  and  gained  15  per  cent. ;  he  now  invested  this 
amount  in  speculation  and  gained  24  per  j3ent.  How 
much  did  he  make  by  the  operation  ? 

Insurance. 
Art    188.  Insurance  is  an  agreement  by  which  a 


AKT.    189.1       STOCKS,    BROKERAGE    AND    COMMISSION.  Ill 

company,  or  individuals,  'obligate  themselves  to  make  good 
any  loss  o:  damage  of  property  by  fire,  shipwreck,  or  other 
casualties. 

The  written  agreement  of  indemnity  issued  by  the  In- 
surers,  sometimes  called  the  underioriters,  to  the  persons 
whose  property  is  insured,  is  called  the  Policy. 

The  insurance  is  effected  in  consideration  of  a  sum  of 
money,  called  a  Premium,  which  is  estimated  at  a  certain 
rate  per  cent,  on  the  amount  insured,  and  is  paid  before- 
hand, to  the  insurers. 

1.  If  A  gets  his  ship  and  cargo  insured  for  $86950, 
from  New  York  to  Liverpool,  at  2  per  cent.;  how  much 
will  be  the  amount  of  the  premium  ? 

2.  An  insurance  of  $18640  was  effected  on  the  ship 
Baltic,  at  2^  per  cent.  How  much  did  the  premium 
amount  to  ? 

3.  A  dwelling,  yalued  at  $1485,  was  insured,  at  |  of  1 
per  cent.     How  much  was  the  premium  ? 

4.  A  steamboat,  valued  at  $55016,  has  an  insurance 
effected  on  |  of  its  value,  at  3|  per  cent.  How  much  is 
the  premium  ? 

5.  A  gentleman  has  his  dwelling  insured  for  $8640,  at 
29  cents  on  $100.     What  is  the  premium  ? 

6.  A  person  at  the  age  of  40,  effects  an  insurance  on 
his  life  for  3  years  for  the  sum  of  $12800,  at  the  rate  of 
$1"95  on  $160  per  annum.  How  much  is  the  annual  pre- 
mium ? 

t.  An  individual,  going  to  California  with  the  intention 
of  returning  at  the  expiration  of  3  years,  effects  an  insu- 
rance of  $9988  on  his  life,  at  i  of  |  of  1^  per  cent,  per 
annum.     How  much  is  the  annual  premium  ? 

Stocks,  Brokerage  and  Commission. 

Art.  189.  Stocks  are  government  funds,  and  the  cap- 
ital of  incorporated  institutions,  such  as  banks,  railroad 
and  manufacturing  companies,  &c.  Stocks  are  divided 
into  shares,  usually  varying  from  $50  to  $500  each,  the 
market  value  of  which  is  at  times  variable. 

8* 


178  PERCENTAGE.  [cHAP.    VIII, 

The  'par  value  of  a  share  is  its  original  cost.  When  it 
sells  for  more  thaa  its  original  cost,  it  is  said  to  be  abi/oe 
par,  or  at  an  advance ;  when  it  sells  for  less,  it  is  below 
par,  or  at  a  discount. 

The  rise  oi  fall  in  stocks  is  computed  at  a  certain  per 
cent,  on  the  par  value  of  the  shares. 

Art.  190.  Brokerage  is  the  percentage  paid  to  bro- 
kers, or  dealers  in  stocks,  money,  bills  of  credit,  and  for 
the  transaction  of  business. 

Art.  191.  Commission  is  the  percentage  paid  to  agents 
and  commission  merchants,  for  the  purchase,  sale,  or  care 
of  property,  and  for  the  transaction  of  other  business. 

The  rate  per  cent,  of  Brokerage  or  Commission,  varies 
in  different  places,  and  depends  upon  the  nature  of  the 
business  transacted. 

1.  What  will  $9864  par  value  of  bank  stock  cost,  at 
18  per  cent,  advance  ? 

Remark.— Find  18  per.  cent,  of  $9864  and  add  it  to  the  $9864  ;  the  sum 
will  be  the  amount  required. 

2.  How  much  must  be  given  for  25  shares  in  the  Hud- 
Boon  River  Railroad,  at  12^  per  cent,  advance,  the  shares 
being  $340  each  ? 

3.  What  is  the  value  of  27  share?  of  canal  stock,  at 
18f  per  cent,  advance,  the  shares  being  $150  each  ? 

4.  How  much  will  be  the  cost  of  18  shares  of  bank 
stock,  at  I7f  per  cent,  below  par,  the  shares  being  $240 
each  ? 

5.  Bought  87  shares  of  a  certain  stock,  at  13^  per  cent, 
below  par,  and  sold  the  same,  at  17f  per  cent,  above  par; 
how  much  did  I  gain,  the  original  shares  being  $184 
each  ? 

6.  A  gentleman  paid  a  broker  f  of  1  per  cent,  to  invest 
$84860  in  government  funds.  How  much  was  the  bro- 
kerage ? 

7.  A  lady,  having  $84847,  paid  an  agent  If  per  cent, 
commission  a  year,  to  take  care  of  it  for  her.  To  how 
much  did  the  ommission  annually  amount? 

8.  An  acent   sells  8484  barrels  of  flour,  at  $5-87i  a 


ART.    192.J  CUSTOM    HOUSE    BUSINESS.  It9 

barrel,  and  charges  If  per  cent,  commission.  How  much 
money  must  he  pay  to  his  employer  after  retaining  his 
commission  ? 

9.  A  merchant,  having  8646  barrels,  gave  an  agent  2f 
per  cent,  commission  for  selling  it.  How  much  did  the 
merchant  receive,  after  deducting  the  commission,  if  it 
were  sold,  at$15'8T^  a  barrel  ? 

10.  A  bank,  failing,  has  in  circulation  $984840,  and  is 
able  to  pay  only  87^  per  cent.  How  much  money  has  the 
bank  on  hand  ? 

11.  A  broker  in  New  York  exchanged  $87846  on  a 
certain  bank  in  Ohio,  for  f  per  cent.  How  much  was  the 
brokerage  ? 

12.  A  merchant  in  Cincinnati  sends  to  a  commission 
merchant  in  New  York  $4536"42  to  lay  out  in  goods,  after 
reserving  bis  commission,  which  was  5  per  cent.  How 
much  was  his  commission  ? 

Solution. — It  will  be  understood  that  the  agent  receives  5 
per  cent,  or  jf^^  =  -^\  of  the  money  laid  out  for  goods  only, 
and  not  of  his  commission ;  therefore,  if  to  gV?  (his  commis- 
sion,) we  add  |^;  (the  money  expended  for  goods.)  we  have 
1^  equal  to  the  sum  of  the  commission  and  amount  paid  for 
the  goods,  which  is  §4536-42.  Hence,  ^V  of  the  money  paid 
for  the  goods,  (which  equals  the  commission,)  is  -^j  of  $4536-42 
=  $20602;  and  ||},- the  amount  paid  for  goods,  is  20  times 
$206-02  =  $4120-40. 

13.  A  farmer  sends  to  a  broker  $84&t2,  to  be  invested 
in  government  funds;  after  deducting  the  brokerage  which 
was,  at  4  per  cent,  on  the  amount  invested.  How  much 
was  invested,  and  how  much  was  the  brokerage  ? 

^14.  A  commission  merchant  receives  $14760  to  pur- 
chase silk,  with  what  remained  after  deducting  his  com- 
mission of  2i  per  cent.  How  many  pieces  of  silk  did  he 
buy,  providing  it  was  $32  a  piece  ? 

Custom  House  Business. 

-    Art.  192.  Duties  are  taxes  levied  by  government  on 
goods  imported. 
These  duties  constitute  the  revenue  of  the  country,  and 


180  PERCENTAGE. 

are  collected  bj  Custom  House  officers,  at  the  ports  of 
entry. 

Duties  are  specific  or  ad  valorem.  A  specific  duty  is  a 
certain  sum  imposed,  on  a  ton,  cwt.,  hogshead,  bushel, 
yard,  &c.,  regardless  of  the  value  of  the  commodity. 

An  ad  valorem  duty  is  a  certain  percentage,  on  the  cost 
of  the  articles  in  the  country  from  which  they  are  im- 
ported. 

Gross  weight  is  the  entire  weight  of  the  commodity, 
together  with  the  cask,  box,  or  bag,  &c.,  containing  it. 

Tare  is  an  allowance  made  for  the  weight  of  the  cask, 
box,  or  bag,  &c.,  containing  the  meichaudise. 

Draft  is  an  allowance  for  waste.  Leakage  is  an  allow- 
ance of  2  per  cent,  for  the  waste  of  liquors  in  transpor- 
tation. 

Net  weight  is  what  remains  after  all  deductions. 

The  usual  allowance  for  draft  is  as  follows: — 


lbs. 

lb. 

lbs.                      lbs. 

On 

112 

1 

From      336  to  1120    4 

From' 

112  to  224 

2 

''       1120  to  2016     7 

u 

244  to  336 

3 

More  than  2016           9 

Note. — The  draft  although  it  is  not  mentioned  in  the  question,  must  be  do- 
ductad.  before  the  other  stated  allowances  are  made 
In  ad  valorem  duties  no  deduction  is  made. 

Art.   193.  To  find  the  specific  duty  on  goods. 

From  the  given  quantity  deduct  all  allowance,  and  multi- 
ply the  remainder  by  the  duty  on  a  unit  of  the  given  quantity. 
The  product  will  he  the  required  duty. 

1.  What  is  the  duty  on  12  barrels  of  sugar,  each  weigh- 
ing 115  pounds  gross,  at  1^  cents  a  pound;  tare  20  per 
cent.  ? 

OPERATION. 

Gross  weight,  2100  lbs. 

Draft  subtracted,  9  lbs. 

2091  lbs. 
20  per  ct.  of  2091  lbs.  tare,  418-2 

Net  weight,         1672^  lbs.  x  '01^  =  $29274,  duty. 


ART.    194.]  ASSESSMENT    OF   TAXES.  181 

2.  What  is  the  duty  on  4  hogsheads  of  sugar,  each 
weighing  1280  lbs.  gross,  at  2f  cents  a  pound  j  tare  14 
per  cent.  ? 

3.  What  is  the  duty  on  420  bags  of  coffee,  each  weighing 
240  pounds,  at  3  cents  a  pound  ;  tare,  3  per  cent.  ? 

4.  What  is  the  duty  on  210  bags  of  coffee,  the  gross 
weight  of  each  bag  being  190  lbs.,  invoiced*  at  5  cents  a 
pound;  the  tare  being  5  per  cent., and  the  duty  25  per 
cent.  ? 

5.  When  there  is  a  duty  on  tea,  of  10  cents  a  pound, 
what  must  be  paid  on  45  chests,  each  weighing  120  lbs.  ; 
tare  10  per  cent.  ? 

6.  At  35  per  cent,  ad  valorem,  what  will  be  the  duty 
on  436  yards  of  satin,  at  $r75  a  yard  ? 

7.  What  is  the  duty  on  85  bags  of  pepper,  each  weighing 
140  lbs.  gross,  invoiced  at  6^  cents  a  pound,  at  3^  per 
cent.  ;  tare  5  per  cent.  ? 

8.  What  is  the  ad  valorem  duty,  at  31i  per  cent.,  on  40 
pieces  of  silk,  each  containing  35  yards,  invoiced  at  $2'25 
a  yard  ? 

9.  What  is  the  duty,  at  18  cents  a  gallon,  on  15  casks 
of  wine,  each  containing  75  gallons  ? 

10.  What  is  the  ad  valorem  duty,  at  62^  per  cent.,  on 
a  case  of  silks,  invoiced  at  $95800  ? 

11.  What  is  the  duty  on  10  barrels  of  Spanish  tobacco, 
each  weighing  145  lbs.  gross;  tare  8  percent.,  at  6|  cents 
a  pound  ? 

12.  What  is  the  duty,  at  40  per.  cent,  ad  valorem,  on  15 
cases  of  French*  broadcloth,  each  case  containing  25  pieces, 
and  each  piece  35  yards,  invoiced  at  $3  95  a  yard  ? 


Assessment  of  Taxes. 

Art.  194.  Taxes  are  moneys  paid  by  the  people,  to 
defray  government  expenses.  Taxes  are  assessed  on  the 
citizens  in  proportion  to  their  real  estate-\  and  personal  pro* 

*  An  invoice  is  a  list  of  the  articles  imported,  and  the  cost  thereof, 
t  Real  Estate  is  immovable  property,  as  lands,  houses,  &c. 


182  PEKCENTAGE.  [cHAP.    VIII. 

'perty,^  except  the  poll-tax,  which  is  so  much  for  each  male 
individual  over  21  years  of  age,  regardless  of  his  property. 

Before  taxes  are  assessed,  an  inventory  of  all  taxable 
property  in  the  state,  county,  or  town  in  which  they  are 
to  be  paid,  must  be  made  ;  together  with  a  list  of  the 
number  of  individuals  liable  to  pay  a  poll-tax. 

Then,  from  the  sum  to  be  raised,  subtract  the  amount 
of  the  poll-taxes,  and  divide  the  remainder  by  the  amount 
of  taxable  property,  which  will  give  the  sura  to  be  paid  on 
$1,  and  multiply  this  sum,, expressed  in  decimals,  by  each 
man's  inventory,  and  the  product  will  be  the  tax  on  his 
property. 

1.  A  tax  of  $840't5  is  to  be  raised  in  a  town  containing 
65  polls.  The  taxable  property  in  the  town  amounts  to 
$4^00.  Each  poll-tax  is  0.75.  What  will  be  A's  tax, 
whose  property  is  valued  at  $375,  and  who  pays  one  poll  t 

Ans.  $6-94  nearly. 

OPERATION. 

$840-75  the  tax  to  be  raised. 
48-75  the  amount  of  poll-taxes. 

$792-00  Remainder. 


$48*75  the  amount  of  poll-taxes. 

mh  =  '0165  the  tax  on  $1. 
375  X  -0165  =  $6- 1875  tax  on  property. 
-75      poll-tax. 

$6-9375  A^mount. 
Explanation. — We  find  the  amount  of  the  poll-taxes  to  be 
65  X  $'75  =  $4875,  which  we  deduct  from  $84075,  and  have 
$792.  If  on  $48000  there  are  $792  taxes  to  be  paid,  on  $1 
there  must  be  paid  j^^^j^  of  $792  =  $00165,  and  on  $375,  A's 
inventory,  375  times  $0165  =  $6-1875.  This  being  increased 
by  1  poll-tax  =  $6-94,  A's  tax. 

Remark. — After  having  determined  the  amount  to  be  paid  on  $1,  the  work 
of  determining  the  tnx  of  each  particular  individual  may  be  facilitated  by 
forming  the  following  table.  If  we  desire  to  find  the  tax  on  $600.  remove  the 
decimal  point  in  the  tax  on  $.5  two  places  to  the  right,  and  we  have  $S  :25,  tha 


♦  Personal  Property  h  that  which  is  movable,  as  money,  furniture,  cattle,  &c 


A.RT, 


195.] 


PROFIT   AND   LOSS. 


183 


tax  on  $500.  The  pupil  will  readily  understand  the  application  of  this  table, 
and  will  also  perceive  that  it  is  the  best  one  that  can  be  formed,  although  not 
the  one  usually  given  by  arithmeticians. 


$ 

S  1      $ 

$ 

Tax  on 

1 

is 

•0165 

Tax  on  11 

8   -1815 

(( 

2 

(( 

•033 

"   12 

"   -198 

a 

3 

u 

•0495 

"   13 

"   ^2145 

(( 

4 

u 

•066 

"   14 

"   -231 

u 

5 

u 

•0825 

"   15 

'   -2475 

u 

6 

C( 

•099 

"   16 

'   -264 

11 

7 

(( 

•1155 

u      17 

"   -2805 

cc 

8 

" 

•132 

"   18 

"   -297 

(I 

9 

u 

•1485 

"   19 

"   -3135 

u 

10 

u 

•165 

2.  By  the  above  table,  what  would  be  the  tax  on  $984, 
there  being  1  poll  ? 

3.  If  I  pay  4  polls,  and  am  worth  |1718-40,  how  much 
is  my  tax  ? 

4.  How  much  is  that  man's  tax,  who  pays  2  polls,   and 
is  worth  $284-86  ? 

5. .  How  much  is  that  man's  tax,  who  pays  3  polls,  and 
is  worth  $8972-50  ? 

6.  How  much  is  that  man's  tax,  who  pays  5  polls,  and 
is  worth  $1784-84? 

7.  How  much  is  that  man's  tax,  who  is  worth  $1984-35, 
and  pays  2  polls  ? 


Profit  and  Loss. 

Art.  195.  Profit  and  Loss  refer  to  the  amount  which 
the  merchant  or  other  business  man,  gains  or  loses  in  busi- 
ness transactions. 

1.  Bought  47  barrels  of  sugar,  at  $14-87i  a  barrel,  and 
sold  it  at  $16-121  a  barrel.     How  much  did  I  gain  ? 
'     2.  Bought  184  cords  of  wood,  at  $8T8f  a  cord,  and 
sold  it,  at  $4 '371  a  cord.     How  much  did  I  gain  by  the 
operation  ? 

3.  Bought  387  barrels  of  flour,  at  $5-93^  a  barrel,  and 
sold  it,  at  $6*7^  a  barrel.     How  much  was  the  gain  ? 

4    Bought  16  barrels-  of  sugar,  each  containing  195 


184  PERCENTAGE.  ( CHAP.  VI^ 

pounds,  at  $13"84  a  barrel,  and  sold  it  for  $.09|  a  pound. 
How  much  was  the  gain  ? 

5.  Bought  flour,  at  $6*20  a  barrd,  and  sold  it  so  as  to 
gain  20  per  cent.  ;  for  how  much  did  I  sell  it  a  barrel  ? 

Solution If  on  100  cents  I  gain  20  cents.,  on  Icent.  I  will 

gain  y|^  of  20  cents  =  ^^,5_  =  |  of  a  cent.  Therefore,  I  gain  ^ 
of  what  it  cost,  }  of  $6-20  =  $1*24,  which  added  to  the  cost 
equals  $7*44,  what  I  must  sell  it  for.     Or, 

Find  20  per  cent,  of  $6-20  ;  thus,  $6-20  X  '20  =  $1-24;  to 
which  add  the  cost,  and  we  have  $7*44,  what  it  must.be  sold 
for  a  barrel. 

6.  Bought  broadcloth,  at  $5*85  a  yard,  and  sold  it  so 
as  to  gain  25  per  cent.;  for  how  much  did  I  sell  it  a 
yard  ? 

t.  A  horse  was  bought  for  $285-^5;  for  how  much 
must  it  be  sold  to  gain  20  per  cent.  ? 

8.  A  merchant  bought  185  barrels  of  pork,  at  $18-95  a 
barrel  ;  but  it  becoming  damaged,  he  was  obliged  to  lose 
35  per  cent,  on  the  sale  of  it.  How  much  did  he  receive 
for  it  all  ? 

9.  A  merchant  bought  25  pieces  of  silk,  each  contain- 
ing 37|  yards,  for  $675'40,  and  sold  it  so  as  to  gain  33^ 
per  cent.     For  how  much  did  he  sell  it  a  yard  ? 

10.  A  quantity  of  butter  was  bought  for  $150,  and  sold 
for  $200 ;  how  much  was  the  gain  per  cent. 

Solution.— On  $150  the  gain  is  $200— $150  =  $50.  If  on 
$150  there  is  a  gain  of  $50.  on  $1,  the  gain  will  be  j\^  of  $50 
=  j^-g\  =  ^  of  a  dollar,  or  33^  per  cent.  1 

11.  A  gentleman  invested  4280  in  speculation,  and  at 
the  end  of  a  year  realized  $5350;  how  much  per  cent,  did 
he  gain  ? 

12.  A  horse  was  bought  for  $240,  and  sold  for  $400; 
how  much  was  the  gain  per  cent  ? 

13.  A  gentleman  sold  a  horse  for  $150,  and  thereby 
gained  25  per  cent.;  how  much  did  the  horse  cost  him  ? 

Solution. — If  he  gained  25  cents  on  100  cents,  on  1  cent,  he 
gained  y^gr  =  j  of  a  cent.  Therefore,  he  gained  \  of  what  the 
horse  cost  him,  which  added  to  |,  the  cost  of  the  horse,  =  |  of 


ART.   195. J  PRACTICAL   QUESTIONS.  185 

the  cost  of  the  horse,  which  is  equal  to  $150,  what  he  sold 
the  horse  for ;  and  ],  =  |  of  $150  =  $30  ]  and  |,  the  cost  of 
the  horse,  =  4  times  $30  =  $120. 

14.  A.  quantity  of  salt  was  sold  for  $864,  which  waa 
331  per  cent,  more  than  it  cost  him;  how  much  did  it  cost 
him? 

15.  If  in  1  year  the  principal  and  interest  of  a  certain 
note,  at  9|  per  cent.,  amount  to  $12000.  How  much  was 
the  face  of  the  note  ? 

16.  A  quantity  of  rye  was  sold  for  $1896,  which  was 
18|  per  cent  more  than  it  cost.     How  much  did  it  cost  ? 

PRACTICAL    QUESTIONS   IN   PROFIT   AND    LOSS. 

1.  If  I  buy  218  yards  of  broadcloth,  at  $4'64  a  yard, 

and  sell  it  at  $6"95i  a  yard;  how  much  do  I  gain  by  the 
operation  ? 

2.  If  I  pay  $846  for  a  quantity  of  wheat ;  for  what 
must  I  sell  it  to  gain  23^  per  cent.  ? 

3.  Sold  149  barrels  of  cider,  at  $4-8tj  a  barrel,  and 
thereby  gained  37|  per  cent.     What  did  it  cost  a  barrel  ? 

4.  Bought  480  gallons  of  molasses,  at  28  cents  a  gallon, 
and  sold  it  for  $168.     How  much  did  I  gain  per  cent. 

5.  A  house  that  cost  $1500,  was  sold  for  $1250.  What 
was  the  loss  per  cent.  ? 

6.  A  farm  that  cost  $6500,  was  sold  for  $9100.  What 
was  the  gain  per  cent.  ? 

7.  Bought  raisins,  at  $3  a  box;  how  much  will  be  the 
loss  per  cent,  if  I  sell  it,  at  $2*50  a  box  ? 

8.  Sold  280  yards  of  cloth  for  $*I00,  and  thereby  gained 
25  per  cent.,  for  how  much  should  I  have  sold  it  a  yard, 
to  lose  20  per  cent.  ? 

9.  If  I  sell  15  yards  of  broadcloth  for  $66,  and  thereby 
gain  10  per  cent.,  how  ought  I  to  have  sold  it  a  yard  to 
have  lost  25  per  cent.  ? 

10.  A  quantity  of  wheat  was  sold  for  $3 60' 90,  which 
was  10  per  cent,  less  than  its  original  cost.;  what  would 
have  been  the  gain  per  cent,  if  it  had  been  sold  for 
$450-15? 


186  •    PERCENTAGE.  [cHAP.    VIII 

11.  Sold  45  boxes  of  damaged  raisins  for  $103-50,  which 
was  at  a  loss  of  8  per  cent. ;  how  should  I  have  sold  them 
a  box  to  have  gained  '3  per  cent.  ? 

12.  A  house  and  lot  was  sold  for  $2t00,  which  was 
8  per  cent,  more  than  its  value;  what  would  have  been 
the  gain  per  cent,  if  it  had  been  sold  for  $28333  ? 

13.  A  mechanic  built  a  house  for  $1980,  which  was  10 
per  cent,  less  than  what  it  was  worth ;  how  much  should  he 
have  received  for  it  so  as  to  have  made  37^  per  cent.  ? 

14.  A  gentleman  sold  two  farms  for  $3680  a  piece;  for 
one  he  received  25  per  cent,  more  than  its  value;  and  for 
the  other,  25  per  cent,  less  than  its  value.  Did  he  gain  or 
lose  by  the  operation,  and  how  much  ? 

15.  A  merchant  sold  two  boxes  of  goods  for  $540  a 
piece;  on  one  he  gained  20  per  cent,  and  on  the  other  he 
lost  20  per  cent.  Did  he  gain  or  lose  by  the  operation, 
afud  how  much  ? 

16.  A  speculator  sold  two  building  lots  for  $1200  a 
piece,  on  one  he  received  37^  #per  cent,  more  than  it  was 
worth,  and  on  the  other  25  per  cent,  less  than  what  it  was 
worth.     Did  he  gain  or  lose,  and  how  much  ? 

SIMPLE  INTEREST. 

Art.  196.  Interest  is  money  due  for  the  use  of  money 
or  its  equivalent ;  and  is  estimated  at  a  certain  rate  "per 
cent,  'per  a7mum,  which  is  generally  fixed  by  law. 

The  Principal  is  the  sum  on  which  the  interest  is  paid. 

The  Amount  is  the  sum  of  the  principal  and  interest. 

By  t.  per  cent,  is  meant  1  cents  on  100  cents,  $7  on  $100, 
or  7  OTi'lOO,  whatever  be  the  denomination. 

The  rate  per  cent,  is  different  in  different  States.  In 
the  State  of  New  York  it  is  *I  per  cent.,  and  in  the  New 
England  States  it  is  6  per  cent.,  &c. 

CASE    I. 

1.  What  is  the  interest  on  $68C  for  6  years,  at  t  per 
cent.  ? 


ART.   196.]  SIMPLE    INTEREST.  187 

OPERATION. 

$•07  int-.  of  $1  for  1  year. 

$•42    «    "    "    "   6  years. 
680 

3360 
252 


$285-60  int.  of  $680  for  6  years,  at  7  per  cent. 

Explanation. — If  the  interest  of  $1  for  1  year  is  7  cents, 
the  interest  for  6  years  will  be  6  times  7  cents,  equal  to  42 
cents.  If  the  interest  of  §1  is  42  cents,  the  interest  of  $680 
is  680  times  $42,  equal  to  $285  60. 

Remark. — Much  care  should  be  taken  to  keep  the  decimal  point  in  its 
proper  place. 

2.  What  is  the  interest  of  $4t0  for  4  years,  at  T  per 
cent.  ? 

3.  What  is  the  interest  of  $683  for  2^  years,  at  6  per 
cent.  ? 

4.  What  is  the  interest  of  $846-4t  for  3f  years,  at  1 
per  cent.  ? 

5.  What  is  the  interest  of  $86*42  for  3^  years,  at  8  per 
cent.  ? 

6.  What  is  the  interest  of  $224-45  for  6|  years,  at  6  per 
cent.  ? 

1.  What  is  the  interest  of  $249'98  for  4f  years,  at  t 
per  cent.  ? 

8.  What  is  the  interest  of  $1*84  for  1  years,  at  5^ 
per  cent.  ? 

9.  What  is  the  interest  of  $163^^  for  3|  years,  at  6^ 
per  cent.  ? 

10.  What  is  the  interest  of  $215-12^  for  4f  years,  at 
8|  per  cent.  ? 

CASE  n. 

To  find  the  interest  on  any  sum  of  money,  for  any  given 
time,  at  6  per  ceE(%. 


188  PERCENTAGE.  [CHAP.    VIII. 


The  interest  of  $1  for  12  months,  (or  1  year.)  is  $00G, 
which  is  equal  to  half  the  number  of  months.  Therefore,  half 
yf  the  number  of  months  equals  the  interest,  in  cents,  of  %\  for 
Ihesame  number  of  montJis.  The  interest  of  $1  for  12  months 
being  $006,  the  interest  for  2  months.  (=  y^^;  or  ^  of  a  year,) 
is  I  X  $006  =  $0-01:  Again,  6  days  is  = /^,  or  ^V  of  2 
months  of  30  days  each  ;  therefore,  the  interest  of  $1  for  6 
days  is  y^  X  $001  =  $0'001.  Therefore,  one-sixth  of  the  num- 
ber of  days  equals  the  interest^  in  mills,  of%lfor  the  same  numbei 
of  days. 

Hence,  to  find  the  interest  of  $1  for  any  given  time,  at  6 
per  cent. : 

Call  half  the  number  of  months  cents,  and  one-sixth  the 
number  of  days,  mills.  The  interest  of  $1  being  found, 
multiply  it  by  the  number  of  dollars  in  the  given  princi- 
pal, and  the  product  will  be  the  interest  requ'red. 

1.  What  is  the  interest  of  $58t*36for  2  years  4  months 
and  24  days  at  6  per  cent.  ?  "" 

operation. 

2  years  4  months  =  28  months.  Calling  the  half  of  the 
28  months  cents,  we  have  ; 

$.14,  int.  of  $1  for  2  years  and  4  months,  at  6  per  ceni. 
Calling  I  of  the  24  days  mills,  we  have  ; 

$004,  int.  of  $1  for  24  days,  at  6  per  cent. 
Hence,  $  -14,      int.  of  $1  for  2  yrs.  4  mo.  at  6  per  cent. 
•004,     u     a     u     u    24  days,  at  6  per  cent. 


$  .144, 

(  gives 
|6per 

the  int.  of 

587-36 

cent. 

864 

432 

1008 

1152 

■720 

for  2  yrs.  4  mo.  24  days,  at 


$84-57984  j  int.  of  $587-36^for  the  given  time  and  the  given 
I  rate  per  cent. 

2.  What  is  the  interest  of  $84-25  for  1  year  and  6 
months,  at  6  per  cent.  ?  * 


iRT.  191.]  SIMPLE   INTEREST.  189 

3.  What  is  the  interest  of  $184'50  for  3  years  and  8 
months,  at  6  per  cent.  ? 

4.  What  is  the  interest  of  $273-84  for  2  years  and  9 
months,  at  6  per  cent.  ? 

5.  What  is  the  interest  of  $84*1-80  for  4  years  1  months 
and  12  days^  at  6  per  cent.  ? 

6.  What  is  the  interest  of  $684-45  for  3  years  8  months 
and  18  days,  at  6  per  cent.  ? 

T.  What  is  the  interest  of  $849-95  for  5  years  5  months 
and  6  days,  at  6  per  cent.  ? 

Remark. — To  find  the  Amount  add  the  principal  and  interest  together. 

8.  What  is  the  amount  of  $684*45  for  2  years  3  months 
and  18  days,  at  6  per  cent.  ? 

9.  What  is  the  amount  of  $483 '85  for  3  years  5  months 
and  24  days,  at  6  per  cent.  ? 

10.  What  is  the  amount  of  $101-01  for  6- years  8 
months  and  14  days,  at  6  per  cent.  ? 

11.  What  is  the  amount  of  $849-&'7i  for  2  years  9 
months  25  days,  at  6  per  cent.  ? 

12.  What  is  the  interest  of  $88*88  for  4  years  11 
months  and  22  days,  at  6  per  cent.  ? 

CASE  III. 

Art.  lOT.  To  find  the  interest  on  any  given  sum  for 
any  given  time,  at  any  given  rate  per  cent.  First,  find 
the  interest  of  $1  for  the  given  time,  at  6  per  cent.,  (See 
Case  2;)  then  take  as  many  sixths  of  the  interest  as  are  equal 
to  the  given  per  cent.,  which  will  be  the  interest  of  $1  for  the 
given  time  and  rate  per  cent.;  then  multiply  this  interest  ly 
the  principal. 

If  the  interest  is  at  t,  9,  or  11  per  cent.,  &c.,  it  is 
evident  that,  if  to  the  interest  of  $1,  at  6  per  cent.,  we  add 
its  J,  I,  or  f,  &c.,  it  will  give  the  interest  of  $1,  at  1,  9,  or 
11,  per  cent.,  &c.,  respectively.  If  the  interest  is  at  2^3, 
or  5  per  cent.,  &c.';  then,  from  the  interest  of  $1,  at  6]per 
cent.,  we  must  take  its  |,  |,  or  \,  &c.,  which  will  give  the 
interest  of  $1,  at  2,  3,  or  5  per  cent.,  &c.,  respectively. 


190  PERCENTAGE.  [CHAP.  VIII 

1.  What  is  the  interest  of  $260  for  1  year  6  months  and 
18  days,  at  8  per  cent.  ? 

OPERATION. 

$•093,  int.  of  $1  for  the  given  time,  at  6  per  cent. 
•031,  ""two-sixths  of  the  above  interest. 

$•124,  int.  of  $1  for  the  given  time,  at  the  given  rate  per  cent 

260 

7440 
248 


$32-240  interest  required. 

2.  What  is  the  interest  of  $84*15  for  2  years  and  10 
months,  at  1  per  cent.  ? 

3.  What  is  the  interest  of  $65' 65  for  1  year  11  months 
and  23  days,  at  1  per  cent.  ? 

4.  What  is  the  interest  of  $384'3ti  for  2  years  and  9 
months  and  16  days,  at  8  per  cent.  ? 

5.  What  is  the  interest  of  $284'95  for  3  years  8  months 
and  20  days,  at  1  per  cent.  ? 

6.  What  is  the  interest  of  $84V3'?^  for  4  years  1 
months,  at  T  per  cent.  ? 

I.  What  is  the  interest  of  $1284'62|  for  2  years  10 
months  and  4  days,  at  7  per  cent.  ? 

8.  What  is  the  interest  of  $884*88  for  4  years  5  months 
and  5  days,  at  5  per  cent.  ? 

9.  What  is  the  interest  of  $841*65  for  5  years  9  months 
and  15  days,  at  5  per  cent.  ? 

10.  What  is  the  interest  of  $8484*84  for  1  years  4 
months  and  20  days,  at  4  per  cent.  ? 

II.  What  is  the  interest  of  $1465*811  for  8  years  8 
months  and  8  days,  at  3  per  cent.  ? 

Rkmark. — If  the  principal  be  given  in  English  money,  reduce  the  shillings, 
pence  and  farthings,  to  the  decimal  of  a  pound  ;  then  proceed  as  it  Federal 
money. 

12.  What  is  the  interest  of  £S4: 10s  Qd.  for  3  years  and 
8  months,  at  1  per  cent.  ? 


ART.    198.]  SIMPLE   INTEREST.  191 

13.  What  is  the  interest  of  £U5  Us.  Sd.  for  2  years 
and  6  months,  at  7  per  cent.  ? 

14.  What  is  the  interest  of  ^284  12^.  lOd.  for  1  year 
8  months  and  12  days,  at  8  per  cent.  ? 

15.  What  is  the  interest  of  ^£384  10.^.  6^^.  for  3  years 
8  months  and  24  days,  at  t  per  cent,  ? 

Art.  198.  The  following  method  of  computing  interest 
avoids  the  use  of  fractions,  and  may,  therefore,  be  preferred 
by  some. 

We  shall  in  accordance  with  general  usage,  reckon  30 
days  to  the  month,  and  12  months  to  the  year. 

1.  What  is  the  interest  of  $460  for  2  years  1  months, 
at  9  per  cent.  ? 

OPERATION. 

$460 
•09 

$41-40,  interest  for  1  year. 
31 

4140 
12420 

12)1283-40 

$106-95  interest  required. 

Explanation. — 1  find  the  interest  of  $460  for  1  year,  (12 
months.)  at  9  per  cent,  to  be  $41*40.  In  the  given  time  there 
are  31  months.  If  the  interest  of  $460  for  12  months  is  $41-40, 
for  1  month  it  is  jL  as  much :  and  for  31  months,  it  is  31  times 
r^  =  ^of:  $41-40  =  $106  95.  Hence,  to  find  the  interest  of 
any  sum,  when  the  time  is  given  in  years  and  months, 

Multiply  the  interest  of  the  principal  for  1  year  hy  the 
number  of  months  and  divide  the  product  iy  l'2i. 

For  a  similar  reason,  when  the  time  is  given  in  years, 
months  and  days, 

Multiply  the.  interest  of  the  principal  for  1  year  hy  the 
number  of  days  and  divide  the  product  by  360,  the  quotient 
will  be  the  interest  required. 


192  PERCENTAGE.  [cHAP.    VIII. 


It  may  be  inferred  from  what  has  already  been  remarked, 
that  none  of  the  preceding  methods  of  computing  interest  is 
strictly  correct  5  however  they  are  in  general  use.  The  follow- 
ing correct  method  is  adopted  by  many  bankers  and  brokers. 

Art.  199.  Muliiply  t/i£  interest  of  the  principal  for  1 
year  by  the  exact  number  of  days  it  has  been  on  interest,  and 
divide  the  product  by  365,  the  quotient  will  be  the  interest 
required. 

2.  What  IS  the  interest  of  $t20  for  2*years  9  months 
and  25  days,  at  8^  per  cent.  ? 

OPERATION. 

$720 
•08^ 


5760 
360 


61 '20  interest  for  1  year. 
2  years  9  months  25  days  =  1015  days. 

30600 
6120 
6120 


360)62118-00($172-55  interest  required. 
360 


2611 
2520 


918 
720 

&c. 

Remark. — The  abov.e  question  is  solved  by  the  method  given  under  Art. 
198.  The  pupil  should  also  solve  the  same  and  the  following  questions  by 
Art.  199,  that  he  may  discover  the  difl'erence  between  the  correct  and  the 
(ncorrect  method  of  calculation. 

3.  What  is  the  interest  of  $14-40  for  3  years  7  months, 
at  7  per  cent,  ? 

4.  What  is  the  interest  of  $25*20  for  4  years  5  months 
and  17  days,  at  9  per  cent.  ? 


ART,  200.] 


SIMPLE    lOTEREST. 


19S 


5.  What  is  the  interest  of  $100'80  for  5  years  9  months 
and  20  days,  at  5  per  cent.  ? 

6.  What  is  the  interest  of  $201*60  for  3  years  1  months 
and  25  days,  at  6  per  cent.  ? 

7.  What  is  the  interest  of  $403-20  for  3  years  8  months 
and  8  days,  at  8^  per  cent.  ? 

8.  What  is  the  intereot  of  $806-40  for  4  years  5  months 
and  21  days,  at  3^  per  cent.  ? 

9.  What  is  the  interest  of  $720  for  6  years  6  months 
and  6  days,  at  5|  per  cent.  ? 

10.  What  is  the  interest  of  $1440  for  1  year  9  months 
and  15  days,  at  8^  per  cent.  ? 

Art.  200.  Many  prefer  to  calculate  interest  by  mulH- 
plying  the  principal  by  the  rate  per  cent.,  and  this  product  by 
the  number  of  years  ;  then  add  the  interest  for  the  months  and 
days,  found  by  raeans  of  aliquot  parts,  to  the  last  product. 

TABLE. 


ALIQUOT  i>ARTS    OF 

A    YEAR    OR    MONTH. 

mo.             yr. 

days.            mo. 

2  =  i^ 

3    =    y 

til 

10    =    ^ 
12    =    f 

18     =    1 

9    =    '( 

10    =     1 

20    =    1 

11     =    Ii 

&c.,  &c. 

Remark. — It  is  customary  in  the  calculation  of  interest,  to  reckon  30  days 
to  the  month,  and  1*2  months  to  the  year,  although  this  is  not  true,  as  some  of 
the  months  contain  more,  and  one  of  them  less  than  30  days  ;  hence,  the  results 
obtained  in  these  calculations  are  sometimes  too  large,  and  at  other  times  too 
small  yet  they  are  sufficiently  correct  for  all  practical  purposes.  But  should 
it  he  desired  to  compute  the  interest  with  more  accuracy,  it  may  be  done  by 
finding  the  number  of  days  the  principal  has  been  on  interest,  by  the  table,  and 
consider  this  number  of  days  as  such  a  part  of  365,  a  year.     {See  Akt.  1'j9.) 

■  1.  What  is  the  interest  of  $240*50  for  3  years  4  months 
and  15  days,  at  8|-  per  cent.  ? 

9 


194 


PERCENTAGE. 


[chap.  VIII 


OPERATION. 


4  mo.  =  ^  yr. 

15  days  =  i  mo. 
or  I  of  4  mo. 


$  240-50 


•OSi 


19-2400 

1-2025 


$20-4425  interest  for  1  year. 
3 


$61-3275  interest  for  3  years. 
6 -81411  interest  for  4  months. 
-8517f  interest  for  15  days. 

$68-9934-}-  interest  required. 


1.  What  is  the  interest  of  $1200'12i  for  6  years  and  4 
months,  at  5  per  cent.  ? 

2.  What  is  the  amount  of  $8t"95  for  2  years  3  montha 
and  20  days,  at  7  per  cent.  ? 

3.  What  is  the  amount  of  $47*84  for  4  years  1  month 
and  25  days,  at  Q^  per  cent.  ? 

4.  What  is  the  amount  of  $144'44  for  3  years  6  months 
and  18  days,  at  7|  per  cent.  ? 

5.  What  is  the  amount  of  $650*30  for  3  years  7  months 
and  12  days,  at  7|  per  cent.  ? 

6.  What  is  the  amount  of  $460*40  for  4  years  8  months 
and  15  days,  at  8f  per  cent.  ? 

7.  What  is  the  interest  of  $640*12i  from  Jan.  24frh, 
1840,  to  March  28th,  1841,  at  6i  per  cent.  ? 

8.  What  is  the  interest  of   $485*9Hf  from  Feb.  5th, 
1842,  to  Aug.  20th,  1844,  at  7^  per  cent.  ? 

9.  What  is  the  interest,  at  5f  per  cent.,  of  $846*84, 
from  Jan.  8th,  until  Nov.  20th  ? 

10.  What  is  the  interest,  at  8|  per  cent.,  of  $384*25 
from  Jan.  12th,  1853,  to  April  4th,  1854  ? 

11.  What  is  the  amount  of  $144*45  from  Aug.  29th 
1852,  to  Nov.  28th,  1853  ? 

12.  What  is  the  interest  of  $1200-121  from  May  22Qd 
1852,  to  Sept.  9th,  1854  ? 

13.  What  is  the  interest,  at  9|  per  cent,  of  $145*60 
from  July  14th,  1851,  to  Sept.  9th,  1853  ?  ' 


ART.    201.]  PROBLEMS   IN   INTEREST.  1&5 

.14.  What  is  the  interest  of  $846-80  from  Sept.  8th, 
1847,  to  Aug.  8th,  1853  ? 

1q.  What  is  the  interest  of  $t84-93f  from  Feb.  2nd, 
1850,  to  April  24th,  1854  ? 

PROBLEMS    IN    INTEREST. 

Art.  201.  The  Principal,  Time,  Rate  per  cent.,  and 
Interest,  have  such  a  relation  to  one  another,  that  any 
three  of  them  being  given,  the  remaining  one  can  readily 
be  found  by  analysis.  , 

Note. — For  a  complete  analysis  of  Interest,  Discount  and  Percentage  of  every 
description,  see  the  last  chapter  in  the  "American  intellectual  Arithmetic." 

Problem  1. — Given  the  rate  per  cent.,  time  and  interest 
to  find  the  principal. 

1.  What  principal  will,  in  2  years  and  6  months,  at  6 
per  cent,  give  $6' 18  interest  ? 

Solution 2  years  and  6  months  equals  |  years.     The  in 

terest  of  $1  for  1  year  is  6  cents,  and  for  J  of  a  year,  ^  of  6 
cents  =  3  cents ;  and  for  4  years,  5  times  3  cents  =  15  cents. 
If  the  interest  on  100  cents  is  15  cents,  on  1  cent  it  is  y|^  of 
15  cts.  =  ^%  =  2^  of  a  cent.  Therefore,  2^,5^  of  the  principal 
equals  the  interest,  which  is  $6*18  ;  and  u\  of  the  principal  = 
i  of  S618  =  $2-06,  and  |^,  the  principar=  20  times  $2-06  = 
$41-20. 

Remark. — A  similar  method  of  analysis  without  further  Uustration  can 
be  readily  applied  by  the  pupil  to  all  the  following  problems. 

The  interest  on  any  sum  is  as  many  times  greater  than 
the  interest  on  $1,  as  that  sum  is  greater  than  $1.  Hence, 
questions  like  the  above  ma,y  be  solved  by  dividing  the 
given  interest  by  the  interest  of  $1,  at  the  given  rate  per  cent, 
for  the  given  time.  „ 

2.  What  principal  will,  in  4  years  and  9  months,  at  8 
per  cent.,  give  $19'38  interest  ? 

3.  What  principal  will,  in  3  years  8  months  and  15 
days,  at  7  per  cent.,  give  $177-551  interest? 

4.  What  principal  will,  in  4  years  9  months  and  18 
days,  at  6  per  cent.,  give  $86-688  interest  ? 


196  PERCENTAGE.  [CHAP.    VIII. 

6.  What  principal  will,  in  10  years  10  months  and  20 
days,  at  6|-  per  cent.,  give  $1411653  interest  ? 

Problem  2. — Given  the  principal,  the  rate  per  cent., 
and  the  interest,  to  find  the  time. 

1.  In  what  time  will  $26,  at  6  per  cent,,  give  $1*95 
interest  ? 

Art.  202.  The  interest  on  a  given  principal  is  in  pro- 
portion to  the  time^  other  things  remaining  the  same. 
Hence,  to  find  the  time,  the  other  three  things  being  given; 
Divide  the  given  interest  by  the  interest  of  the  given  jprindjpal, 
at  the  given  rate  'per  cent,  for  1  year;  or  solve  it  by 
Analysis. 

2^In  what  time  will  $300,  at  8  per  cent.,  give  $20 
interest  ? 

3.  In  what  time  will  $90-25,  at  6  per  cent.,  give  $4't5 
interest  ? 

4.  In  what  time  will  $284"75,  at  5f  per  cent.,  give 
$18-^5  interest? 

5.  In  what  time  will  $114'95,  at  t^  per  cent.,  give 
$34-8ti  interest? 

Problem  3. — Given  the  principal,  the  tim€,  and  the 
interest,  to  find  the  rate  per  cent  ? 

1.  The  interest  of  $65,  for  10  months  is  $3  25.  What 
is  the  rate  per  cent.  ? 

Art.  203.  The  interest  on  a  given  principal  is  in  pro- 
portion to  the  rate  per  cent.,  other  things  remaining  the 
same.  Hence,  to  find  the  rate  per  cent.,  the  remaining 
three  things  being  given ;  Divide  the  given  interest  by  the 
interest  of  the  given  principal,  at  1  per  cent.,  for  the  given 
time. 

2.  The  interest  of  $120  for  2  years  9  months  and  12 
days,  is  $13"36.     What  is  the  rate  per  cent.  ? 

3.  The  interest  of  $3t5  for  3  years  and  6  months,  is 
$9t'125.     What  is  Ihe  rate  per  cent.  ? 


AKT.    205.]  DISCOUNT.  19T 

4.  The  interest  of  $248  for  2  years  1  month  and  20 
days,  is  $29-194.     What  is  the  rate  per  cent.  ? 

5.  The  interest  of  $184-85  for  two  years  8  months  and 
18  days,  is  $31-84.-    What  is  the  rate  per  cent.  ? 

Problem  4. — Given  the  amount,  time,  and  rate  per  cent., 
to  find  the  principal  ? 

1.  What  principal  will,  in  4  years  6  mouths,  at  8  per 
cent.,  amount  to  $430  ? 

Art.  204.  The  amount  of  different  principals,  for  the 
same  time,  and  at  the  same  rate  per  cent.,  are  to  each 
other  as  those  principals.  Hence,  Dividing  the  given 
amount  by  the  amount  of  $1,  at  the  given  rate  per  ant.,  for 
the  given  time,  will  give  the  principal. 

2.  What  principal  will,  in  t  years  and  6  months,  at  8 
per  cent.,  amount  to  $2600  ? 

8.  What  principal  will,  in  2  years  and  4  months,  at  6 
per  cent.,  amount  to  $640  ? 

4.  What  principal  will,  in  5  years  8^  months,  at  1  per 
cent.,  amount  to  $2100  ? 

5.  What  principal  will,  in  4  years  4. months,  at  6  per 
cent,,  amount  to  $3800  ? 

DISCOUNT. 

Art.  205.  Discount  is  an  allowance,  according  to  the 
rate  per  cent.,  made  for  the  payment  of  money  before  it  is 
due. 

The  present  worth  of  a  debt,  payable  at  some  future  time, 
without  interest,  is  such  a  sum  as  will,  in  the  given  time, 
and  at  the  given  rate  per  cent,,  amount  to  the  debt. 
Hence,  the  present  worth  of  any  sum  of  money,  payable  at 
some  future  time,  without  interest,  is  equal  to  the  quotient 
arising  from  dividing  that  sum  hy  the  amount  o/  $1,  at  the 
given  rate  per  cent.,  for  the  given  time. 

The  Discount  equals  the  amount,  mm\xs,i\\Q present  worth. 

1.  What  is  the  present  worth  of  $644,  due  4  years  9 
months  and  18  days  hence,  at  6  per  cent.? 


198  PERCENTAGE.  [CHAP.    VIII. 

ExPLANA  noN. — $1-288  is  the  amount  of  $1  for  the  given  time, 
and  the  given  rate  per  cent.  Now  we  have  the  proportion 
$1288,  amount:  $644,  amount::  $1,  present  worth  :  presgni 
worth,  required.  This,  solved  gives  $500  for  the  required  pre- 
sent worth. 

2.  What  is  the  present  worth  of  $840,  due  3  years  and 
4  months  hence,  at  6  per  cent.  ? 

3.  What  is  the  present  worth  of  $1140,  due  2i  years 
hence,  at  6  per  cent.  ? 

4.  What  is  the  discount  on  $450,  due  2  years  and  9 
months  hence,  at  t  per  cent.  ? 

5.  What  is  the  discount  on  $1200,  due  3  years,  4 
months  hence,  at  4f  per  cent  ? 

6.  What  is  the  discount  on  $84*25,  due  3  years  8  months 
and  24  days  hence,  at  8  per  cent.  ? 

t  What  is  the  present  worth  of  $9632,  due  1  year  8 
months  and  12  days  hence,  at  6  per  cent.  ? 

8.  What  is  the  present  worth  of  $52  32,  due  6  years  1 
month  18  days  hence,  at  6  per  cent.  ? 

9.  What  is  the  discount  on  $464*80,  due  3  years  8  months 
and  15  days  hence,  at  7  per  cent.  ? 

10.  Bought  $984-45  worth  of  goods  on  a  credit  of  9 
months.  How  much  money  would  discharge  the  debt,  at 
the  time  of  receiving  the  goods,  interest  being  9  per  cent.  ? 

11.  A  merchant  bought  goods  to  the  amount  of  $3328: 
1  of  \i)  was  on  a  credit  of  6  months,  and  the  remainder  on 
a  credit  of  9  months.  How  much  money  would  discharge 
the  debt,  interest  being  8|  per  cent.  ? 

12.  A  merchant  bought  goods  to  the  amouut  of  $2480: 
$812  of  which  was  on  a  credit  of  3  months;  $832,  on  a 
credit  of  8  months;  and  the  remainder  on  a  credit  of  9 
months.  How  much  ready  money  would  discharge  the 
debt,  interest  being  6  per  cent.  ? 

13.  A  merchant  bought  goods  to  the  amount  of  $1600: 
i  of  which  was  on  a  credit  of  3  months;  ^  on  a  credit  of 
9  months ;  and  the  remainder  on  a  credit  of  1  year.  How 
much  ready  money  would  discharge  the  debt,  interest  being 
8  per  cent.  ? 


*rt.  206.]  partial  paiments.  199 

Partial  Payments. 

Art,  206.  Partial  Payvients  are  payments,  or  indorse 
ments,*  made  at  various  times,  of  a  part  of  a  note,  hond^ 
or  obligation. 

Tiie  method  adopted  by  the  Supreme  Court  of  th~e  United 
States  for  the  calculation  of  interest  on  notes,  and  other 
obligations,  where  partial  payments  have  been  made,  is  as 
follows  ; — 

"  Apply  the  payment,  in  the  first  place,  to  the  discharge  of 
the  interest  then  due.  If  the  payment  exceed  the  interest,  tht 
surplus  goes  towards  discharging  the  principal,  and  the  sub- 
seque7Lt  interest  is  to  be  computed  on  the  balance  of  the  principal 
rtmaining  due.  If  the  payment  be  less  than  the  interest,  the 
surplus  of  interest  must  not  be  taken  to  augment  the  principal ; 
hut  interest  co7itinues  on  the  former  principal  until  the  period 
ivhen  the  payments  taken  together  exceed  the  interest  due,  and 
then  the  surplus  is  to  be  applied  towards  discharging  the 
principal ;  and  interest  is  to  be  computed  on  the  balance,  as 
aforesaid." 


1800.  Bethany,  Sept.  8th,  1850. 


[1.]  On  demand,  I  promise  to  pay  Thomas  Brooking, 
or  bearer,  eight  hundred  and  sixty  dollars,  with  interest. 
Value  received.  John  Jackson. 

On  this  note  are  the  following  indorsements  : — 

Nov.  20th,  1851,  received  $382-24. 
May    8th,  1853,       "        $23845. 

How  much  is  due  Dec.  29th,  1853,  allowing  t  per  cent, 
interest  ? 

Rkmark.— It  will  be  of  some  assistance  to  the  pupil  to  arrange  the  date  of  the 
note,  the  payments,  time  of  settlement,  and  the  intervals  of  time  between  pay- 
ments, together  with  the  interest  of  $1  for  the  given  time,  at  6  per  cent.,  as 
follows  : — 


♦  Deriv  ?si  from  a  Latin  phrase  signifying  "  upon  the  back  ;"  as  the  paymwits 
are  written  across  the  back  of  the  note. 


200  PERCENTAGE.  [cHAP.    VIIX. 

Intervals 

of  time.  Int.  (f$l,at 

years,    mo.  da.      y.  mo.  aa      raymer^ts.  6  per  cent. 

Date  of  note,     .     .  1850    9    8 

1st  payment,       .      1851  11  20  12  12  $382-24      |0-072 

2(i  payment,      .     .  1853    5     8  1  5  18  238-45       0-088 

Time  of  settlement,  1853  12  29  0  7  21  00385 

Face  of  the  note,  or  principal,         .  .  .     $86000 

Int.  on  the  same,  at  7  per  cent.,  to  Nov.  20th;  1851,      7224 

Amount  due  on  the  note,       .  "        "        "      $932-24 

First  payment,         .....      382-24 


Amount  remaining  due, — 2nd  principal,           .  $550-00 

Int.  on  the  same,  from  Nov.  20, 1851.  to  May  8, 1853,  56-46 

Amount  due,  May  8th,  1851,          .            .            .  $606-46 

Second  payment,            ....  238-45 

Amount  remaining  due,— 3rd  principal,     .            .  $36801f 

Int.  from  May  8th,  1853,  to  the  time  of  settlement, '  16 -534- 

Amount  due  Dec.  29th,  1853.             .            .  $384-54| 


$786//o.  New  York,  Jan.  13th,  1848. 

[2.]  On  demand,  I  promise  to  pay  to  the  order  of 
JEenry  Morton,  seven  hundred  and  thirty-six  dollars,  with 
interest,  at  7,  per  cent.     Yalue  received. 

Kace  B.  Bonhoovan. 

On  this  note  are  the  following  indorsements  :— 

Received  Oct.  7th,    1849,  $275-45. 

"      Aug.  25th,  1850,  $386-38. 

How  much  remains  due  Sept.  19th,  1852  ? 


$684yVo-  Cincinnati,  July  26th,  1849. 

[3.]  On  demand,  I  promise  to  pay  James  Benort,  or 
bearer,  six  hundred  and  eighty-four  dollars,  with  interest, 
at  6  per  cent.    Value  received.        John  P.  Trumbal. 


ART.    208.] 


ANNUAL   INTEREST. 


201 


On  this  note  are  the  following  indorsements. 

Received  Jan.  20th,  1850,  $284-75. 
March  14th,  1851,  $84-75. 
July  26th,       1853,  $384-37|. 

How  much  remains  due  Sept.  8th,  1854  ? 


ANNUAL  INTEREST. 

In  the  computation  o^  Annual  Interest,  the  yearly  increase 
of  the  principal  is  equal  to  the  annual  interest  of  the  first 
principal. 

If  a  note  or  obligation  for  |500  should  be  made  payable 
in  5  years  with  annual  interest  at  6  per  cent.,  and  another 
for  $500,  payable  in  1  year  with  annual  interest  at  6  per 
cent.,  and  no  payment  whatever  should  be  made  on  either 
until  the  expiration  of  5  years,  the  amount  of  the  obligations 
would  be  equal. 

1.  What  is  the  annual  interest  of  $520  for  3  years,  at  5 
per  cent.  1 

OPERATION. 

First  Principal,        -        $520 


-05 


"      Interest, 

Second  Principal, 
"        Interest, 

Third  Principal, 
"      Interest, 


$26-00 

$520 
26 

$546 
•05 

$27-30 

$546 
26 

$572 
•05 

$28-60 


First  year's  interest,  $26*00 


Second  year's  interest,  $2Y"80 


Third  year's  interest,  $2860 


Annual  interest  of  $520  for  3  years,        -        -        -        $81-90 

2.  What  is  the  annual  interest  of  $814  for  5  years,  at  6 
per  cent.  ? 

3.  What  is  the  annual  interest  of  $840  for  6  years,  at  7 
per  cent.  ? 


202  COMPOUND    INTEREST.  ^  [CHAP.  VIIL 

COMPOUND  INTEREST. 

Compound  Interest  is  interest  on  both  principal  and  in- 
terest together;  the  interest,  annually  or  semi-annually,  be- 
ing successively  taken  to  augment  the  principab 

1.  What  is  the  compound  interest  of  $520  for  3  years, 
at  5  per  cent.  1 


Principal, 
Per  cent. 

$520 
•05 

Interest  for  1  year, 

$2600 
520 

Amount  for  1  year  or  2nd  principal, 

$546 
•05 

Interest  of  $546,  for  1  year. 

-      $27-30 
546 

Amount  the  2nd  year,  or  3rd  principal 

,       $573-30 
-05 

Interest  of  $573-30  for  1  year. 

$28-6650 
$573-30 

Amount  the  3rd  year,              '    - 
Original  principal, 

$601-965 
$520 

Compound  interest  required,  -  $8r96|- 

2.  What  is  the  compound  interest  of  $384*50  for  3  years, 
at  8  per  cent.  *? 

3.  What  is  the  compound  interest  of  |840,  for  2  years, 
mterest  payable  semi-annually,  at  8  per  cent.  ?  «i 

4.  What  is  the  compound  interest  of  $460,  for  3  years^ 
interest  payable  half  yearly,  at  6  per  cent.  ? 

5.  What  is  the  difference  between  the  annual  and  com 
pound  interest  of  $850  for  8  years,  at  6  per  cent.  1 

Banking  and  Notes. 

Art.  209.  A  Bank  is.  an  institution  created  bylaw 
for  the  purpose  of  issuing  bank  notes,  or  bank  bills,  which 
circulate  as  money,  and  are  redeemable  in  specie,  on  pre- 
sentation to  the  bank;  also  for  loaning  money,  receiving 
deposits,  and  dealing  in  exchange. 

The  Capital  Stpck  of  the  bank,  is  divided  into  shares 


ART.    209. j  FORMS    OF    NOTES.  203 

which  are  owned  by  various  individuals  called  stock- 
holders. 

The  Stockholders,  annually  elect  a  board  of  Directors, 
to  manage  the  concerns  of  the  Bank.  This  board  elects  a 
Cashier^  and  one  of  their  number  as  President  of  the  bank. 
The  President  and  Cashier  sign  all  bills  issued  by  the 
Bank. 

A  promisory  note  is  a  positive  engagement  in  writing, 
to  pay  a  certain  sum  at  a  specified  time,  to  a  person  desig- 
nated in  the  note,  or  to  his  order,  or  to  the  bearer. 

Forms  of  Notes. 
[No.  1.] 


$T8yVo-  Liberty,  July  3d,  1849. 

On  demand,  I  promise  to  pay  Edward  Fox,  seventy- 
eight  and  yVo  dollars,  with  interest.     Yalue  received. 

Edward  Everts. 

[No.  2.] 

Negotiable  Note. 


-j-^^.  Kingston,  Aug.  25th,  1850. 


U4JL 


One  year  after  date,  I  promise  to  pay  to  the  order  of 
Moses  Morton,  ninety-nitie  and  yVo  dollars,  with  interest. 
Value  received.  John  Frontz. 

[No.  3.] 
Note  Payable  to  Bearer. 


$365tVo-  Ellenville,  Sept.  10th,  1851. 

Six  months  after  date,  I  promise  to  pay  Isaac  Ingraham, 
or  bearer,  three  hundred  sixty-five  and  y^^  dollars,  with 
interest.     Yalue  received.  Simeon  Sa.wyer. 

[No.  4.] 
Note  Payable  at  a  Bank. 


$47yVo.  Buffalo,  Oct.  15th,  1851 

Forty  days  after  date,  I  promise  to  pay  to  the  order  of 


204  PERCENTAGE.  [cHAP.    VIII. 

Joseph  Langhorn,  at  the  Union  Bank,  in  Sullivan  Co., 
N.  Y.,  forty-seven  and  jW  dollars,  with  interest.  Yalue 
received.  Henry  Mifflin. 

The  r^  rawer  or  Maker  of  a  note  is  the  person  who 
signs  it. 

Note  No.  1,  can  be  collected  by  Edward  Fox  only, 
therefore,  it  is  not  negotiable. 

Note  No.  2,  becomes  collectable  by  any  person  holding 
it,  after  Moses  Morton  writes  his  name  on  the  back  of  it, 
which  is  called  indorsing  the  note.  Moses  Morton  is  now 
called  the  Indorser.  When  this  note  becomes  due  pay- 
ment must  be  demanded  of  the  Drawer,  and  if  he  refuses 
or  neglects  to  pay  it,  notice  must  be  given  without  delay 
to  the  Indorser,  demanding  payment  of  him. 

The  payment  of  note  No.  3,  can  be  demanded  by  any 
person  holding  it;  the  Drawer  alone  is  responsible. 

If  Joseph  Langhorn  writes  his  name  on  the  back  of 
note  No.  4,  payment  may  then  be  demanded  of  him,  if 
the  drawer  refuses  or  neglects  to  pay  it  at  the  specified 
time. 

A  note  that  has  not  the  words  "  Value  received  ^^  on  it, 
is  invalid.  ^ 

Bank  Discount. 

Art.  210.  Bank  Discount  is  the  sum  paid  to  a  bank 
for  the  payment  of  a  note  before  it  becomes  due. 

The  amount  named  in  a  note,  is  called  the  face  of  the 
note.  The  discount  is  the  interest  on  the  face  of  the  note 
for  3  days  more  than  the  time  specified,  and  is  paid  in  ad- 
vance.* These  3  days  are  called  days  of  grace,  as  the  bor- 
rower is  not  obliged  to  make  payment  until  their  expira- 
tion.    Hence,  to  compute  bank  discount. 

Find  the  interest  on  the  face  of  the  note  for  3  days  more 
than  the  time  specif  ed  ;  this  will  he  the  discount.  From  the 
face  of  the  note,  deduct  the  discount,  and  the  remainder  will 
he  the  present  value  of  the  note. 

*  Taking  interest  in  advance  is  usurious,  and  has  been  discontinued  by 
many  banks  ;  and  instead  'liereof,  they  deduct  the  true  discount,  found  by 
Articlo  205. 


ART.  211.]  BANK   DISCOUNT.  205 

1.  What  is  the  bank  discount  on  $240  for  6  months,  at 
T  per  cent.  ? 

2.  What  is  the  bank  discount  ou  $460  for  4  months,  at 
8  per  cent.  ? 

3.  What  is  the  bank  discount  on  $150-50  for  2  months 
and  15  days,  at  6  per  cent.  ? 

4.  What  is  the  bank  discount  on  $4t5'85  for  3  months, 
at  1  per  cent.  ? 

5.  What  is  the  bank  discount  of  a  note  of  $8t5'50,  for 
8  months  21  days,  at  t  percent.  ? 

6.  What  sum  must  a  bank  pay  for  a  note  of  $385*^5, 
payable  in  6  months,  discount,  at  7  per  cent.  ? 

7.  What  is  the  present  value  of  a  note  of  $875"25,  dis- 
counted at  a  bank  for  8  months  and  9  days,  at  6  per  cent.  ? 

8.  What  is  the  present  value  of  a  note  of  $84650,  dis- 
counted at  a  bank  for  4  months  and  15  days,  at  7  per 
cent.  ? 

9.  What  is  the  present  value  of  a  note  of  $8484*50  dis- 
counted at  a  bank  for  7  months  and  9  days  at  7  per  cent.  ? 

Art.  211.  GiYen  the  present  value  of  a,  hankMe  note, 
the  rate  per  cent.,  and  the  time  for  which  it  is  to  be  dis- 
counted, to  find  the  face  of  the  note. 

1.  What  must  be  the  face  of  a  bankable  note  so  that 
when  discounted  for  4  months  and  15  days,  at  6  per  cent., 
it  shall  give  a  present  value  of  $1954  ? 

Solution. — The  discount  of  $1  for  the  given  time,  at  the 
given  rate  per  cent.,  is  $023;  hence,  $l_$-023=$-977,  the 
the  present  value  of  §1  for  the  given  time,  at  the  given  rate 
per  cent. 

We  now  have  the  proportion,  $-977,  present  value  :  $1954, 
present  value  :  :  $1  amount, :  required  amount,  which  is  $2000, 
the  face  of  the  note. 

Hence,  we  infer  in  general,  that  if  we  Divide  the  given 
PRESENT  VALUE  by  the  PRESENT  VALUE  of  $1  foT  the  given 
time  and  at  the  given  rate  per  cent.,  hank  discount  ;  the  quo- 
tient will  he  the  amount  or  face  of  the  note. 

2.  What  must  be  the  face  of  a  bankable  note,  so  that 
when  discounted  for  6  months  and  10  days,  at  6  per  cent., 
it  will  gire  $85,  present  value  ? 


206  AVERAGE.  [chap.    VIII 

3.  What  must  be  the  amount  of  a  banlfable  note,  so 
that  when  discounted  for  4  months  and  21  days,  at  1  per 
cent ,  it  shall  give  $84 '95  present  value  ? 

4.  What  must  be  the  amount  of  a  bankable  note  so  that 
when  discounted  for  4  months  and  9  days,  at  8  per  cent., 
the  borrower  shall  receive  $384  ? 

5.  What  must  be  the  amount  of  a  bankable  note,  so 
that  when  discounted  for  6  months  and  27  days,  at  t  oer 
cent.,  the  borrower  shall  receive  $580  ? 


AVERAGE. 

Art.  2-12.  If  the  sum  of  a  series  o^  'promiscuous  quan- 
tities, be  divided  by  the  number  of  quantities,  the  quotient 
will  be  one  of  a  series  of  equal  quantities,  whose  sum  will 
equal  the  sum  of  the  former  series.  This  quotient  is  called 
the  AVERAGE  of  thc  given  quantities. 

1.  What  is  the  average  of  12,  16,  and  20  ? 

2.  During  six  successive  months  a  laborer  saved  $12, 
$18,  $25,  $30,  $20,  and  $27  a  month  respectively.  How 
many  dollars  did  he  average  a  month  ? 

3.  A  locomotive  made  4  successive  trips  over  a  track 
20  miles  in  length,  in  the  following  times  :  30  minutes  25 
seconds  ;  25  minutes  15  seconds  ;  33  minutes  10  seconds  ; 
and  24  minutes  30  seconds.  What  was  the  average  time 
of  1  trip,  also  of  running  1  mile  ? 

MERCANTILE  CALCULATIONS. 

Equation  of  Payments. 

Art.  213.  Equation  of  Payments  is  the  process  of 
finding  the  average  time  for  the  payment  of  several  sums 
due  at  different  times,  without  loss  to  either  party. 

The  rules  applying  to  mercantile  calculations  will  be  given 
for  the  accommodation  of  book-keepers. 

Art.  214.  The  equated  time  for  the  payment  of  any 
sum,  when  parts  of  it  are  payable  at  different  times,  may 
be  found  by  the  following 


ART.    214.]  MERCANTILE    CALCULATIOXS.  20"7 


Multiply  each  jpaymejit  by  the  time  that  must  elapse  before 
it  becomes  due ;  then  divide  the,  sum  of  these  products  bf-4he 
sum  of  the  payments.  The  quotient  will  be  the  average 
time  required. 

1.  I  purchased  goods  to  the  amount  of  $1200  ;  $300 
of  which  I  am  to  pay  in  2  months  ;  $400  in  3  months  ; 
and  $500  in  6  months.  How  long  a  credit  ought  I  to 
receive,  if  I  pay  the  whole  at  once  ? 


Explanation. — A  credit  on  $300  for  2  months 
is  the  same  as  the  credit  on  $1  for  600  months. 

A  credit  on  $400  for  3  months  is  the  same  as 
the  credit  on  $1  for  1200  months. 

A  credit  on  $500  for  6  months  is  the  same  as 
the  credit  on  $1  for  3000  months. 

Tijerefore.  on  the  whole  sum,  $1900, 1  should 
receive  the  same  as  the  creditor  the  interest  on 
$1  for  4S00  months;  the  $1200  wiU  give  the  same 
interest  in  one-twelve  hundredth  of  4800  month."? 
=4  months,  the  time  in  which  the  whole  amount 
averages  due. 


OPERATION. 

$        mo. 

mo. 

300x2  = 

=    600 

400x3  = 

=  1200 

500x6  = 

=  3000 

1200) 

4800(4  mo. 
4800 

0 


2.  If  I  owe  $900  ;  $200  of  which  is  due  in  2  months  ; 
$300  in  4  months;  and  the  remainder  in  6  months.  What  is 
the  average  time  for  the  payment  of  the  whole  ? 

3.  If  you  owe  a  man  $150,  payable  in  2  months  ;  $260 
payable  in  4  months  ;  $490  payal)le  in  8  months;  at  what 
time  may  you  in  equity  pay  the  whole  ? 

4.  A  merchant  bought  goods  to  the  amount  of  $400,  on 
a  credit  of  4  months;  another  quantity  for  $500,  on  a  credit 
of  5  mo.;  and  another  quantity  for  $800,  on  a  credit  of  8 
mo.  What  is  the  average  time  for  the  payment  of  the  whole  ? 

5.  A  gentleman  owes  a  certain  sum  of  money;  ^  of  which 
is  due  in  3  months;  |  in  4  months;  }  in  12  months.  What 
is  the  average  time  of  payment  ? 

Remark — We  will  suppose  the  amount  owed  is  $1,  as  it  can  make  no  diffe- 
vence  what  that  amount  is,  since  certain  fractional  parts  of  it  become  due  at 
particular  times. 

6.  A  merchant  bougbi  goods  to  the  amount  of  $3000  ; 


208  AVERAGE.  [chap.  VIII. 

I  of  wliicli  he  paid  in  cash  at  the  time  of  receiving  the 
goods  ;  I  is  to  be  paid  in  6  months;  and  the  remainder  in 
1  year  and  3  months.  What  is  the  average  time  for  the 
payment  of  the  whole  ? 

7.  A  man  purchased  a  farm  for  $3200,  and  agreed  to 
pay  $500  of  it  at  the  expiration  of  3  months  ;  $1200  at 
the  expiration  of  9  months ;  and  the  remainder  in  12  months. 
What  is  the  equated  time  for  the  payment  of  the  whole  ? 

Art.  215.  In  mercantile  transactions  it  is  customary  to 
give  a  credit  of  from  3  to  9  months,  on  bills  of  sale.' 

Art.  216.  To  determine  the  average  time  of  payment 
of  several  sales,  on  different  terms  of  credit,  we  have  the 
following 

RULE. 

Multiply  the  amount  of  each  sale  by  t/ie  tiTue  intervening 
betwee7i  tJie  date  on  which  the  first  amount  falls  due,  and  the 
date  on  which  each  sum  falls  due.  Then  divide  the  sum 
of  these  products  by  the  whole  amoimt  of  debt,  and  the  qivotient 
will  be  the  averaged  time  of  payment,  to  be  counted  forward 
from  the  date  of  the  first  amount  falling  due. 

1.  Purchased  goods  of  Stiiwell,  Brown  &  Co.,  at  dif- 
ferent dates  and  on  different  terms  of  credit ;  as  below 
stated. 

JFeb.       2, 1853,  a  bill  amounting  to  ^460  on  3  months'  credit. 

Feb.        5,    "  "  "  $680  on  4 

March  28,    "  "  "  $560  on  5       " 

April    12,    "  "  "  i840  on  5 

I  wish  to  make  one  payment  of  the  whole  debt.  When, 
per  average,  will  it  become  due  ? 

Solution. — The  above  bills  come  due  respectively  as  follows : 

days.  days.  " 

Due  May  2,  $460  X  00  =  00000 
"  June  5,  $680  X  34  =  23120 
"  Aug.  28,  $560  X  118  =  66080 
"  Sept.  12,  $840  X  133  =  111720 

$2540)       200920(79  days. 
&c.  &c. 


ART.  216.]  MERCANTILE    OALCULATIONS.  209 

The  $460  become  due  May  2nd,  1853;  the  $680  become 
due  34  days  from  May  2nd;  the  $560,  118  days  from 
May  2nd;  and  the  $840,  133  days  from  May  2nd.  By 
equation  of  payments  I  find  these  bills  will  average  due  in 
19  days  from  May  2nd,  which  is  July  20th,  1853. 


If  it  were  required  to  know  how  much  money  would 
balance  the  account  any  time  previous  to  July  20thj  as  April 
15th,  it  is  evident  that  the  present  worth  of  $2540  from  April 
15th  to  July  20th,  would  be  the  sum  required.  * 

When  the  different  sales  are  made  on  the  same  terms 
of  credit,  the  ave]*a.ge  time  for  the  payment  of  the  whole 
debt,  may  be  found  as  taught  by  the  following  question  : 

2.  A  merchant  sold  goods  to  one  of  his  customers,  at 
different  dates;  as  belov^  stated  : 

April    8, 1853,  a  bill  amounting  to  $470  on  6  month's  credit. 
May    17,     "  "  "  $840  on  6 

June  23,     "  ♦'  ««  #980  on  6       «' 

July  10,     «.«  *'  ♦'  ^580  on  6       " 

What  is  the  average  time  for  the  paymei^  of  the  above' 
bills  ? 

OPERATION. 

April  8,  1853,  $470  x  00  =  00000 
May  17,  "  $840  X  39  =  32760 
June  23,  "  $980  X  76  =  74480 
July  10,  «   $580  X  93  =  53940 

$2870)     161180(5^  days. 

From  a  little  reflection  the  pupil  will  discover  that  the  above 
bills  will  average  due  in  56  days  from  the  time  the  first  falls 
due,  which  is  Dec.  3,  1853. 

3,  A  merchant  sold  to  one  of  his  customers  several 
parcels  of  goods,  at  sundry  times,  and  on  different  terms 
of  credit;  as  follows  : 

Feb.        1,  a  bill  amounting  to  ^300  on  4  months'  credit. 
March    7,       "  "  $185  on  5       "  " 

April    15,       "  ««  $280  on  4       " 

May      20,       "  «  $210  on  3       " 

What  is  the  equated  time  for  the  payment  of  all  these  bills  ? 


210  AVERAGE.  [chap.   VIII, 

4.  Purchased  goods  of  a  merchant  at  different  times, 
and  on  different  terms  of  credit  as  follows  : 

March    3,  1853,  a  bill  amounting  to  $847 '10  on  3  months'  credit. 
April      5,     "  "  "  $'645.60  on  4       " 

May      10,     "  "  "  :||!584-75  on  6 

June     15      "  «'  "  $475-84  on  8       '« 

Aug.    18      "  "  "  P84-.95  on  4       " 

What  is  the  equated  time  for  the  payment  of  the  above 
bills  ? 

5.  Purchased  goods  at  sundry  times,  and  on  different 
terms  of  credit,  as  follows  : 

June  4,  1853,  a  bill  amounting  to  ^485-90    on  8  month's  credit. 

July  12,     "  "  "  $675-25    on  4       " 

Aug.  15,     "  "  "  i;81212^on5       «' 

Sept.  22,     "  "  "  $895-25    on  6      ♦' 

Nov.  20,     "  "  '*  $896-70    on  4       " 

What  is  the  equated  time  for  the  payment  of  all  these  bills  ? 

6.  Bought  goods  of  J.  B.  Smith  &  Co.,  at  sundry  times, 
as  shown  by  the  statement  annexed. 

March   2,  1853,  a  bill  amounting  to  $684"20  on  6 months'  credit. 

March  28,  "  ♦'  "  $875-54     " 

April   10,    "  "  "  $484-40     " 

May     20,  "  "  "  $795-45     '«         ««  " 

June    30,    '4         "  "  $840-60     *' 

How  much  money  will  balance  the  account,  July  4, 
1854? 

t.  Bought  goods  of  C.  B.  Hill  &  Co.,  at  different  times, 
and  on  different  terms  of  credit,  as  shown  by  the  state- 
ment annexed  ? 

June-12,  1853,  a  bill  amounting  to  $340-65  on  5  months'  credit. 

July    8,       "  "  "  $595-75     " 

Aug.  10,      "  "  "  $784-85        6 

Oct.    14,      "  "  "  $987-90        8 

Nov.  15,      "  "  "  $878-98        4 

Dec.    19,      "  ♦'  "  $999-99        2 

How  much  money  will  balance  the  account,  Jan.  20, 
1854  ? 

8.  Bought  goods  of  R,  Lancaster  &  Co.  as  shown  by 
the  statement  annexed; 


ART.  211.]  MERCANTILE    CALCULATIONS.  211 

May  4, 1853,  a  bill  amounting  to  ^432-95  on  3  months'  credit. 

May   18,    "         "        .     "  $843-45       2 

Jnne20,    "         "  "  .$732-46       6 

July    8,     "         "  "  $-846-75       7 

Aug.  20,    "         "  "  ^784-78       6 

Sep.   24,    "         "  "  $976-34       4 

What  is  the  equated  time  for  the  payment  of  all  these 
bills  ;  and  how  much  money  would  balance  the  account, 
Nov.  12,  1854? 

Art.  217".  The  rule  given  for  the  Equation  of  Pay- 
ments is  the  one  usually  adopted  by  merchants,  although 
not  strictly  correct,  still  it  is  sufficiently  accurate  for  all 
practical  purposes,  when  small  sums  and  short  periods  of 
time  are  considered. 

This  inaccuracy  will  become  evident  by  inspecting  the 
following  example  : — 

A  owes  B  $4480  ;  $2240  of  which  is  due  in  2  years,  and 
the  remainder  in  10  years.  What  is  the  equated  time  for 
the  payment  of  the  whole,  interest  6  per  cent.  ? 

The  average  time  of  payment  found  by  the  rule  above 
referred  to,  is  6  years.  From  which  we  observe  that 
$2240  is  not  paid  until  4  years  after  it  is  due,  also  that 
$2240  is  paid  4  years  before  it  is  due  ; — these  two  condi- 
tions are  considered  to  mutually  counter-balance  each  other, 
although,  they  do  not.  It  is  evident,  that,  for  deferring 
the  payment  of  the  first  $2240  for  4  years,  A  should  pay  the 
amount  of  $224:0  ioY  the  same  time,  which  is  $2777-60  ; 
but  for  the  remainder,  which  he  pay^  4  years  before  it  is 
due,  he  should  pay  the  present  worth  of  $2240  for  the 
same  time,  which  is  $1806"45.  Hence  the  Rule  occasions 
an  error  of  $277760  +  $1806-45— $4480=$10405. 

Justice  demands  that  interest  should  be  required  on  all 
sums  from  the  time  they  become  due  until  the  payment  is 
made,  and  the  present  worth  of  all  sums  paid  before  thej 
become  due.     Hence  the  following  accurate 

RULE. 

Find  the  present  worth  of  each  of  the  given  amounts  due, 
then,  find  in  what  time  the  sum  of  these  present  worths  will 
amo^mt  to  the  sum  of  all  the  payments. 


212  AVERAGE.  [chap.    VIII. 

Art.  218.  When  a  debt  due  at  some  future  period 
has  received  partial  payments  before  the  time  due,  to  find 
how  long  after  this  time,  the  remainder  may  in  equity  re- 
main unpaid. 

RULE. 

Divide  the  sum  of  the  'products  of  each  payment  into  th( 
time  it  was  paid  before  due^  by  the  sum  remaining  unpaid 
The  quotient  will  be  the  required  time. 

1.  A  owes  $1200,  due  in  6  months;  five  months  before" 
it  is  due  $200  is  paid;  and  3  months  before  it  is  due,  $300 
is  paid.     How  long  after  the  expiration  of  the  6  months 
may  the  remaining  $500  in  equity  remain  unpaid  ? 

operation. 

mo.         mo.  (  Explanation. — A  credit  on  $200  for 

$200  X  5  =  1000  }  5  months  is  the  same  as  a  credit  on 

/  $1  for  1000  months. 

!A  credit  on   $500  for  3  months  is 
the  same  as  a  credit  on  $1  for  1500 
months. 
$500)  2500  ^    , 

The  money  paid  in  advance  affords 

5  mo.  a  profit  equal  to  the  interest  of  $1 

for  2500 ;  the  balance  remaining  due,  $500,  will  afford  the 
same  interest  in  one-five  hundredths  of  2500  months,  which  is 
5  months. 

2.  A  person  owes  $400,  due  at  the  epd  of  10  months. 
At  the  end  of  4  months  he  pays  $100;  3  months  after 
that  he  pays  $50.  How  long  after  the  expiration  of  the 
10  months  may  the  balance  remain  unpaid  ? 

3.  A  merchant  lends  to  a  farmer  $1600,  payable  in  12 
months.  At  the  end  of  4  months  $200  of  it  is  paid  ;  3 
months  after  that  $400  more  is  paid  ;  and  1  month  before 
the  expiration  of  the  12  months  $200  more  is  paid.  How 
long  after  the  expiration  of  the  12  months  may  the  balance 
in  equity  remain  unpaid  ? 

4.  A  lends  B  $1200  for  6  months  ;  at  another  time 
$1800  for  8  months.  For  how  long  a  time  ought  B  to 
lend  A  $2t00,  to  balance  the  favor  ? 


IRT.    219.]  MERCANTILE    CALCULATIONS.  213 

5  I  borrowed  of  my  neighbor  $900  for  5  months  ;  at 
another  time  $800  for  9  months.  For  how  long  a  time 
ought  I  to  lend  my  neighbor  $850  to  balance  the  favor  ? 

Compound  Equation  of  Payments. 
Art.  219.  Compound  Equation  of  Payments  teaches  tho 
method  of  ascertaining  the  time  on  which  the  balance  of 
an  account  that  contains  debit  and  credit  becomes  due  ; 
having  first  learned,  by  Rule  under  Art.  216,  when  the 
debit  and  credit  of  said  account  falls  due,  respectively, 
without  regard  to  their  relation  to  each  other. 


1st.  Multiply  the  number  of  days  between  the  dates  of 
equated  time  by  the  amount  that  first  falls  due  ;  and  divide 
this  product  by  the  difference  between  the  debit  and  credit  of 
the  account  ;  the  quotient  will  be  the  tdae  for  consideration. 

2d.  If  the  larger  amount  comes  due  first,  the  time  is  counted 
BACK  from  the  latest  date  ;  but  if  the  smaller  amount  comes 
due  first,  the  time  is  counted  forward  from  the  latest  date. 

1.  By  equation  of  payments  it  is  found  that  A^s  account 
with  B.  is  as  follows  : 

Dr.  Cr. 

Due  June  4th,      .     .     $400  1  Due  July  24th,     .     .     $900 

When  will  the  balance  of  the  account  become  due  ? 

operation 
Amount  of  Cr.  $900  due  July  24th. 
"  Dr.  $400  due  June  4th. 

Balance  $500  From  June  4th  to  July  24th,  is  50  days. 

days. 

50 
400 


500)20000 

40  days  from.  July  24th,  which  is  Sept.  2d,  the  balance 
becomes  due. 

Explanation — It  is  evident  that  B  should  receive  the  mterest 
on  $400  from  June  4th- to  /uly  24th.    Therefore  B  should  retain 


214  AVERAGE.  1_CHAP.    Vm 

the  balance  ($500)  sufficiently  long  after  it  becomes  due,  to  re- 
ceive the  same  amount  of  interest  on  it  as  he  would  have  re- 
ceived on  the  $400  from  June  4th  to  July  24th. 

If  $400  in  50  days  give  a  certain  interest,  $1  will  give  the 
same  interest  in  400  times  50  days  =  20000  days;  and  $500 
(the  balance)  will  give  the  same  interest  in  -^  of  20000  days 
=40  days;  consequently,  in  40  days  from  July  24th,  the  balance 
becomes  due. 

2.  Suppose  the  above  account  to  stand  as  follows  ? 

Dr.  Cr. 

Due  June  4,     .     .     .     $900  |  Due  July  24,    .     .     .     $400 

At  what  time  must  a  note  for  the  balance  be  dated,  to 
balance  the  account  ? 

Solution. — It  is  evident  that  B  should  receive  the  interest 
on  $900  from  June  4th  to  July  24th,  Therefore,  to  balance 
the  account,  B  should  receive  a  note  of  $500  (the  balance),  of 
such  a  date  that  the  interest  on  it  should,  on  the  24th  day  of 
July,  equal  the  interest  on  $900  for  50  days,  the  time  from 
June  4th  to  July  24th. 

If  $900  in  50  days  give  a  certain  interest,  $1  will  give  the 
(Same  interest  in  900  times  50  days  =  45000  days ;  and  $500 
will  give  the  same  amount  of  interest  in  -^l-^  of  45000  days. 
Hence,  90  days  previous  to  July  24th,  which  is  April  25th,  the 
note  should  be  dated. 

3.  B  has  with  C  the  following  account : — 

1853.  Dr.    I    1854.  Cr. 

Nov.  12.  Due      .     .     $840.  |  Jan.  20.  Due     .     . 


When  will  the  balance  of  the  account  become  due  ? 

4.  C  has  with  D  the  following  account : — 

1853.  Dr.    I    1853.  Cr. 

July  20.  Due      .  - .     $987.  |  Sept.  4.  Due       .     .     $507 

At  what  time  must  a  note  of  the  balance  be  dated  U 
balance  the  account  ? 

5.  At  what  time  will  the  balance  of  the  following  account 
become  due  ? 

1853.  Dr.     I    1854.  Cr. 

Oct.  26.  Due      .     .     $1280.  |  Jan.  16.  Due      .     .     $840. 

6.  When  will  the  balance  of  the  following   account 
become  due  ? 


ART.  220.]  MERCANTILE    CALCULATIONS.  215 

1853.  Dr.    1    1853.  Cr. 

April  21.  Due     .     .     $845  |  June  15.  Due      .     .     $1685 

Cash  Balance. 

Art.  220.  To  find  the  cash  balance  of  an  account 
consisting  of  various  items  of  debit  and  credit,  of  different 
dates,  at  any  specified  time. 

RULE. 

Place  on  the  debtor  or  credit  side,  such  a  sum,  (which  may 
he  called  merchandise  balance, J  as  will  balance  the  account. 

Multiply  the  number  of  dollars  in  each  entry  hy  the  num^ 
her  of  days  from  the  time  the  entry  was  made  to  the  time  of 
settlement ;  and  the  merchandise  balance  by  the  number  of  days 
for  which  credit  was  given.  Then  midti'ply  the  difference 
between  the  sum  of  the  debit,  and  the  sum  of  the  credit  products 
by  the  interest  of  %1  for  1  day ;  this  product  will  he  the 
\nterest  balance. 

When  the  sum  of  the  debit  products  exceed  the  sum  of  the 
redit  products,  the  interest  balance  is  in  favor  of  the  debit 
side  ;  hut  when  the  sum  of  the  credit  products  exceed  the  sum 
of  tJie  debit  products,  it  is  in  favor  of  the  credit  side.  Now 
to  the  merchandise  balance  add  the  interest  balance,  or  subtract 
it,  as  the  case  may  require,  and  you  obtain  the  cash  balance. 

1.  A  has  with  B  the  following  account : — 


1849.  Dr. 

Jan.      2.  To  merchandise  $200 
April  20.  "  "  400 


1849.  Cr. 

Feb.  ^0.  By  merchandise  $100 
May  10.  "  "  .   300 


If  interest  is  estimated  at  t  per  cent.,  and  a  credit  of  60 
days  is  allowed  on  the  different  ^ums,  what  is  the  cash 
balance  August  20,  1849  ? 

Explanation. — Without  interest,  the  cash  balance  would  be 

$200. 

If  no  credit  had  been  given,  the  debits  should  be  increased 
by  the  interest  of  $200  for  230  days,  at  7  per  cent. ;  and  the 
interest  of  $400  for  122  days,  at  7  per  cent.  The  credits  should 
be  increased  by  the  interest  of  $100  for  181  day,  at  7  per  cent. ; 
and  the  interest  of  $300  for  102  day^.  at  7  per  cent. 

Since  a  credit  of  60  days  is  given  on  all  sums,  it  is  evident 


2ie 


AVERAGE. 


[chap.    VIII. 


by  the  above  calculation,  that  we  shall  increase  the  debits  by 
the  interest  of  the  sum  "of  the  debits,  $600,  for  60  days  more 
than  justice  requires.  Also,  that  we  should  increase  the  credits 
by  the  interest  of  the  sum  of  the  credits,  $400,  for  60  days  more 
than  we  should  do. 

Now,  instead  of  deducting  these  items  of  interest  from  the 
amount  of  debit  and  credit  interests,  it  is  plain,  that  it  will  be 
more  convenient  and  equally  just,  to  diminish  the  debit  inter- 
est by  the  interest  of  the  merchandise  balance  for  60  days,  which 
can  be  most  readily  accomplished  by  adding  the  interest  on 
the  merchandise  balance  for  60  days,  to  the  credit  items  of 
interest. 

From  which  we  discover  that  the  interest  balance  is  equal  to 
the  difference  between  the  sum  of  the  debit  interests,  and  the 
sum  of  the  credit  interests  increased  by  the  interest  of  the  mer- 
chandise balance  for  the  time  for  which  credit  was  given. 


DEBITS. 

$  Days. 

200  X  230  =  46000 
400  X  122  =  48800 

94800 
60700 


OPERATION. 

CREDITS. 
$  Days. 

100  X  181  =  18100 

300  X  102  =  30600 

Balance,  200  x  60  =  12000 


60700 


0^ 
365 


X  34100  =r  $6-54  Interest  balance,  nearly. 


Therefore,  the  foregoing  account  becomes  balanced  as 
follows  : — 


1849.  Dr. 

Jan.      2.  To  merchandise,       $200-00 
April  20.     "  "  400  00 

Aug.  20.     "  balance  of  interest,    6-64 


$606-54 


1849. 
Feb.  20.  By  merchandise. 
May  10.     " 
Aug.  20.    "  balance, 


Cr. 

$10000 

300  00 

206  64 

$60654 


Aug.  20.    "  Cash  balance,        $-206  64 

Note. — It  is  customary  in  practice,  when  the  number  of  cents  in  any  of  the 
ptries,  are  less  than  60.  to  omit  them,  and  to  add  $1  when  they  are  80  or  more. 

2.  A  has  with  B  the  following  account : — 


1852.  Dr. 

Jan.     8.  To  merchandise,  $400 
April  24. "  "  800 


1852.  Cr. 

Feb.  10.  By  merchandise,    $300 
May  24.    "  "  500 


TRADE    AND    BARTER-  21 1 

If  interest  is  estimated,  at  T  per  cent,  and  a  credit  of 
60  days  is  allowed  on  the  different  sums,  what  is  the  cash 
balance  Sept.  25th,  1852  ? 

3.  B  has  with  C  the  following  account  : — 


1853. 

Ih: 

1853. 

Cr. 

Feb.  12.  To  merchandise, 

$840 

March  16.  By  merchandise, 

$640 

July  25.    " 

980 

May  14.      ^           - 

780 

Aug.  14.    " 

C40 

Sept.  20.      ♦^            •• 

430 

I 


If  interest  is  estimated  at  8  per  cent.,  and  a  credit  of 
90  days  is  allowed  on  the  different  sums,  what  is  the  cash 
balance  Jan.  10th,  1854  ? 


TEADE    AND   BARTER. 

Art.  221.  Trade  and  Barter  is  the  exchange  of  one 
commodity  for  another  without  loss  to  either  party. 

1.  How  many  yards  of  muslin,  at  $'12^  a  yard,  must 
be  given  for  380  lbs.  of  butter,  at  $'16  a  pound  ? 

2.  How  many  bushels  of  rye,  at  $93^  a  bushel,  must 
be  given  for  187|  lbs.  of  tea,  at  $'62^  a  pound  ? 

3.  A  merchant  exchanged  630  yards  of  cloth,  for  15 
hogsheads  of  wine,  at  $1*10  a  gallon.  How  much  was 
the  cloth  a  yard  ? 

4.  A  farmer  gave  20^  cwt.  of  hops,  at  $6*80  per  cwt., 
for  8  cwt.  3  qrs.  20  lbs.  of  sugar,  and  $80  in  money;  at 
how  much  was  the  sugar  valued  per  pound  ? 

5.  A  farmer  received  for  25  cwt.  2  qrs.  22  pounds  of 
cheese,  at  $081  a  pound,  18  yards  of  cloth,  at  $250  a 
yard;  16  yards  of  muslin,  at  5^  cents  a  yard;  5  pair  of 
boots,  at  $2-75  a  pair;  85  gallons  of  molasses,  at  %'l^^  a 
gallon,  and  the  balance  in  sugar,  at  $'09^  per  pound. 
How  many  pounds  of  sugar  did  he  buy  ? 

6.  A  grocer  barters  860  bushels  of  oats,  which  cost  htm 
$•25  a  bushel,  at  $-37i  a  bushel,  for  cloth  that  cost  $2-86| 
a  yard.  What  is  the  bartering  price  of  the  cloth,  and 
how  many  yards  did  the  grocer  receive  ? 

1.  A  farmer  has  184^  bushels  of  rye,  which  is  worth 
$'84  per  bushel;  but  in  barter  he  is  willing  to  put  it  at 

10 


218  TRADE    AND    BARTER.  ^CHAP.    VIII. 

$•56  a  bushel,  providing  his  neighbor  will  let  him  have 
wheat  worth  $1-24  per  bushel  for  $-91.  Will  he  gain  or 
lose  by  the  bargain,  and  what  per  cent.  ? 

8.  Two  farmers  bartered:  A  had  240  bushels  of  wheat, 
at  $1-50  per  bushel,  for  which  B  gave  him  200  bush,  of 
corn,  at  $'65  per  bush.,  and  the  balance  in  buckwheat,  at 
$•80  a  bushel.  How  much  buckwheat  did  A  receive  of  B  ? 

9.  A  farmer  has  380  bushels  of  wheat,  worth  $1^20  a 
bushel;  but  in  barter  he  will  have  $144  a  bushel.  A 
merchant  has  broadcloth  worth  $.3^60  a  yard ;  and  linen 
worth  $1'40;  at  what  price  per  yard  ought  the  merchant 
to  rate  his  broadcloth  and  linen  to  be  equivalent  to  the 
farmer's  bartering  price,  and  how  many  yards  will  the 
farmer  receive  for  his  wheat,  providing  he  takes  an  equal 
number  of  yards  of  each  ? 

10.  A  and  B  barter:  A  has  560  bushels  of  wheat  worth 
$1^20,  but  in  barter  he  will  have  $1^60  a  bushel;  B  has 
broadcloth  worth  $4^20  a  yard;  how  must  B  sell  his 
broadcloth  a  yard  in  proportion  to  A's  bartering  price  for 
his  wheat,  and  how  many  yards  are  equal  in  value  to  A's 
wheat  ? 

11.  A  had  450  yds.  of  cloth,  worth  $1^20  a  yard,  which 
he  bartered  with  B,  at  $1^45  a  yard;  taking  flour,  at 
$^•50  a  barrel,  which  is  worth  but  $6.  How  much  flour 
will  pay  for  the  cloth;  and  who  gets  the  best  of  the  bar- 
gain ? 

12.  A  farmer  sold  to  a  merchant  one  yoke  of  oxen  for 
$125;  184  bushels  of  corn,  at  $-3Ti  a  bushel;  45  bushels 
of  wheat,  at  $-93f  a  bushel,  April  14th,  1853.  In  pay- 
ment he  received  125  lbs.  of  raisins,  at  lOf  centsa  pound; 
584  pounds  of  sugar,  at  9-]-  cents  a  pound;  and  54  gal- 
lons of  molasses,  at  13|  cents  a  gallon,  Nov.  8th,  1853 
How  much  remains  due;  interest  6  per  cent.  ? 

13.  A  farmer  took  to  market  26 tO  lbs.  of  wheat,  worth 
.$'93f  a  bushel;  and  in  payment  takes  $5^1 1|  to  pay  his 
taxes;  and  for  the  remainder  he  is  to  receive  an  equal 
number  of  yards  of  muslin,  at  *J\  cents  a  yard;  bleached 
muslin,  at  12^  cents;  calico,  at  16|  cents;  and  linen,  at 
31^  cents.     How  many  yards  of  each  did  he  buy  ? 


ART.  221.]  TRADE    AND    BaRTER.  219 

.  14.  A  farmer,  Mr.  Smith,  lent  his  neighbcr,  April  1st, 
16  busliels  of  superior  wheat,  on  condition  that  it  should 
be  paid  in  wheat  of  equal  quality,  on  the  first  of  the 
following  November,  after  adding  3  per  cent,  to  it  for  its 
nse.  The  wheat  his  neighbor  returned  was  7  per  cent, 
inferior  to  that  which  he  received.  How  many  bushels  of 
wheat  should  Mr.  Smith  in  equity  receive  ? 

15.  A  farmer,  Mr.  Jackson,  owes  a  merchant  $560,  May 
1st,  1853.  On  the  above,  Mr.  J.  paid,  Aug.  4th,  1853, 
4f  bushels  of  clover-seed,  at  $6-75  a  bushel,  and  a  yoke 
of  oxen  for  $95.  On  Nov.  15th,  1853,  Mr.  J.  paid  63^ 
bushels  of  rye,  at  $-93f  a  bushel,  and  the  remainder  in 
wheat,  at  $ri2i  a  bushel.  How  many  bushels  did  it  take, 
interest  8  per  cent.  ? 

16.  A  Mr.  Judson  sold  to  Mr.  Wilson,  April  10,  1852  ; 
4  cows,  at  $2375  each  •  15  bushels  of  oats,  at  $"37^  a 
bushel  ;  24  cwt.  3  qrs.  of  hay,  at  $1075  a  ton  ;  and  1 
wagon,  at  $84'95.  Mr.  Wilson,  in  payment,  sold  Mr.  J., 
March  15th,  1853,  3  plows,  at  $6  37^  each;  2  cultivators, 
each  $5-18f  ;  12^  yards  of  broadcloth,  at  $4-85  per  yard  ; 
2  barrels  of  sugar,  at  $17  75  a  barrel  ;  5  sacks  of  salt,  at 
$3*12i  a  sack.  Allowing  7  per  cent,  interest,  how  does  the 
account  stand,  Oct.  12th,  1853. 

17.  A  speculator,  Mr.  Manning,  bought  of  Mr.  Bron- 
8on  a  house  and  lot  for  $2400,  January  1st,  1853,  |  of 
which  was  payable  at  the  time  the  purchase  was  made,  and 
the  remainder  was  to  be  paid,  with  interest  at  7  per  cent, 
in  3  equal  payments  ;  the  first  in  4  months;  the  second  in 
8  months;  and  the  third  in  12  months.  Mr.  B.  sold  Mr.  M. 
485|  bushels  of  potatoes,  at  $'62|-  a  bushel,  April  15th, 
1853  ;  and  August  12th,  1853,  748  bushels  of  corn,  at 
$•47^  a  bushel.  They  are  desirous  of  settling,  Nov.  18th, 
1853.  How  much  in  equity  should  Mr.  Manning  pay  Mr. 
Bronson  ? 

18.  Mr.  Mathews  sold  to  Messrs.  Arnold  &  Co.,  May  12, 
?853,  465  lbs.  of  pork,  at  $'11^  a  pound,  and  a  span  of 
horses  and  pleasure  wagon  for  $684*50,  on  6  months'  credit. 
June  15,  1853,  Mr.  M.  bought  of  Messrs.  A.  &  Co.  47-75 
acres  of  land,  at  $23'25  an  acre,  on  3  months'  credit.    Mr 


TRADE    AVD    BARTEB.  lCHAP.    VIII. 

M.  sold  to  Messrs.  A.  &  Co.  184  bushels  of  wheat,  at  $-93f 
a  busiiel,  for  which  no  credit  is  allowed.  They  settled 
accounts  Nov.  15th,  1853.  Which  was  in  debt  to  the  other, 
and  how  much  ? 

19.  A  farmer  sold  to  a  merchant  41|-  bushels  of  corn,  at 
$-65  a  bushel  ;  84^  bushels  of  rye,  at  $'87^  a  bushel  ;  36f 
bushels  of  buckwheat,  at  $"93f  a  bushel  ;  and  in  payment 
received  12^  lbs.  of  tea,  at  $ri2i  a  pound  ;  15^  pounds 
of  coffee,  at  $16|  a  pound  ;  135  pounds  of  sugar,  at  $-9| 
a  pound  ;  25  gallons  of  molasses,  at  35|-  a  gallon  ;  36|- 
yards  of  linen,  at  $'15|-  a  yard;  25^  yards  of  calico,  at  $16| 
a  yard  ;  24^  yards  of  broadcloth,  at  $3'85^  a  yard  ;  15 
pair  of  shoes,  at  $2*54  a  pair  ;  and  a  set  of  spoons,  knives, 
and  forks,  for  $19' 84.  Which  owed  the  other,  and  how 
much? 

20.  A  farmer  sold  a  mechanic,  April  1st,  1853,  a  span 
of  horses  for  $284;  a  yoke  of  oxen  for  $184;  148^  bushels 
of  grain,  at  $-93f  a  bushel;  and  4  cows,  each  $25-75,  on  a 
credit  of  9  months.  The  mechanic  sold  the  farmer,  May  1, 
1853,  a  lumber  wagon  for  $184;  a  pleasure  wagon  for 
$325;  and  4  plows,  each  $6-75,  on  a  credit  of  3  months. 
They  settle  accounts  Sept.  1st,  1853;  which  is  in  debt, 
and  how  much,  interest  6  per  cent.  ? 

21.  A  farmer  sold  to  a  merchant,  Jan.  3d,  1852,  1764 
pounds  of  pork,  at  8|  cents  a  pound;  1683  pounds  of  beef, 
4f  cents  a  pound;  847  pounds  of  ham,  at  10^  cts.  a  pound; 
March  12th,  1852,  485  bushels  of  oats,  at  $"43|  a  bushel; 
184  bushels  of  rye,  at  $-62i  a  bushel;  284  bushels  of  wheat, 
at  $-93f  a  bushel;  and  487  pounds  of  cheese,  at  11^  cents 
a  pound.  The  farmer  received  of  the  merchant,  March 
25th,  1853,  merchandise  to  the  amount  of  $684-75;  June 
15th,  1853,  merchandise  to  the  amount  of  $84645.  In 
•the  above  transaction  6  months'  credit  was  given  on  all 
the  articles.  Balance  the  account  July  3d,  1853,  at  6 
oer  cent,  interest. 

22.  Mr.  Smith  bought  of  a  speculator  a  farm  containing 
o72  acres,  at  $40  an  acre,  and  was  to  pay  for  it  in  ten  years 
as  follows  :  |  of  the  whole  the  first  year;  |  of  the  remain- 
der the  second  year;  |  of  the  remainder  the  third  yearj 


ART.  222.]  ARITHMETICAL   PROGRESSION.  221 

i  of  the  remainder  the  fourth  year;  ^  of  the  remainder  the 
fifth  year;  i  of  the  remainder  the  sixth  year:  and  the 
remainder  in  four  equal  and  annual  payments,  without 
interest.  The  first  payment  was  made  in  wheat,  at  $1*12^ 
a  bushel;  the  second  in  wheat,  at  $125  a  bushel;  the 
third  in  wheat,  at  $  93f  a  bushel;  the  fourth  in  wheat,  at 
$'95  a  bushel;  the  fifth  in  wheat,  at  $1-10  a  bushel;  the 
sixth  in  wheat,  at  $r20  a  bushel;  and  the  remainder  iu 
wheat,  at  $1"15  a  bushel.  What  was  the  amount  of  each 
payment,  and  the  number  of  bushels  of  wheat  paid  annu- 
ally ?  If  no  payment  had  been  made  until  the  end  of  the 
ten  years,  how  many  bushels  of  wheat,  at  $r35  a  bushel, 
would  have  balanced  the  account,  interest,  at  1  per  cent.  ? 


CH'APTER  IX. 
PROGRESSION 


Arithmetical  Progression. 

Art.  222.  A  series  of  numbers  that  increase  or  de- 
crease by  a  constant  difference,  is  said  to  be  in  Arithmetical 
Progression.  When  the  terms  are  constantly  increasing, 
the  series  is  called  an  Ascending  Arithmetical  Progression  ; 
as,  1,  3,  6,  t,  9,  11,  13,  15,  &c. 

When  they  are  constantly  decreasing,  the  series  is  called 
a  Descending  Arithmetical  Progression  ;  as, 

45,  43,  41,  39,  3t,  Zb,  33,  31,  &c. 

The  first  and  last  terms  of  a  Progression  are  called  the 
extremes,  and  the  other  terms  are  called  the  means. 

From  the  nature  of  an  Arithmetical  Progression,  it  fol- 
lows, that  the  sum  of  the  extremes  is  equal  to  the  sum  of 
any  other  two  terms  equally  distant  from  them,  or  to  twice 
the  middle  term ;  if  the  number  of  terms  is  unequal.    This 


222     ,  PROGRESSION.  [CHAP.    IX. 

wil]  appear  more  plain  by  iuspecting  the  following  Pro- 
gression : — 1,  3,  5,  1,  9,  11,  13,  15,  and  17. 

1,    3,    5,    7,    9  ^  Here  the  sum  of  the  extremes  and 

17,  15,  13,  11,    9  !  the  terms   equally  distant  from  them, 
—  —  —  —  —  I  are  added,  and  found  to  be  equal,   aa 

18,  18,  18,  18,  18  J  above  stated. 

In  Arithmetical  Progression  there  are  five  distinct 
terms  to  be  considered : — 

a,  The  first  term; 

1,    The  last  term; 

n,  The  number  of  terms; 

d,  The  common  difference;  and 

s,    The  sum  of  all  the  terms. 

These  terms  are  so  related  that  any  three  of  them  bemg 
known,  the  remaining  two  may  be  found.  Since  there  are 
five  terms,  and  only  three  of  them  necessary  to  be  known, 
to  find  a  fourth,  it  follows  that  there  may  be  twenty  dis- 
tinct cases  in  Arithmetical  Progression.  We  shall,  how- 
ever, notice  but  few  of  them,  and  refer  the  student  to 
Algebra  for  the  others. 

CASE    I. 

Art.  223.  Given  the  first  term,  the  common  difference, 
and  the  number  of  terms,  to  find  the  last  term. 

1.  What  is  the  25th  term  of  an  arithmetical  progres- 
sion, the  first  term  of  which  is  6  and  the  common  dif- 
ference 4  ? 

Explanation. — The  second  term  of  an  arithmetical  progres- 
sion ascending  is  equal  to  the  first  term  plus  the  common  differ- 
ence ;  the  third  term  is  equal  to  the  first  term  plus  twice  the 
common  difierence  :  and  so  on.  Therefore,  when  we  have  given 
the  first  term,  the  common  difierence  and  the  number  of  terms, 
the  last  term  is  found  by  Adding  the  first  term  to  the  product  of 
the  common  difierence  into  the  number  of  terms  less  one. 

2.  A  man  bought  25  acres  of  land,  giving  $2  for  the 
first  acre,  $8  for  the  second,  $14  for  the  third,  and  so  on, 


ART.  224.]  ARITHMETICAL   PROGRESSION.  223 

iigreasiug  iif  arithmetical  progression.      What  did   the 
last  acre  cost  at  tj^is  rate  ? 

3.  x\  merchant  bought  18  pieces  of  cloth,  giving  $3  for 
the  first  piece;  $5  for  the  second;  $7  for  third,  and  so  on, 

Jncreasiiig  in  arithmetical  progression.  What,  at  this  rate, 
did  the  last  piece  cost  ? 

4.  A  tapering  board,  3^  inches  wide  at  the  narrow  end, 
and  14  feet  long,  is  found  to  increase  in  width  1|  inches 
for  every  foot  in  length.  What  is  the  width  of  the  wide 
end? 

5.  In  a  certain  orchard  there  are  34  rows;  in  the  first 
row  there  are  20  trees;  in  the  second  24;  in  the  third 
28;  and  so  on,  the  number  of  trees  in  each  row  continu- 
ing to  increase  in  an  arithmetical  ratio.  How  many  trees, 
at  this  rate,  are  there  in  the  last  row  ? 

CASE  II. 

Art.  224.  Given  the  first  terra,  the  last  term,  and 
the  number  of  terms,  to  find  the  sum  of  all  the  terms. 

1.  The  first  term  of  an  arithmetical  progression  is  5, 
the  last  term  is  85,  and  the  number  of  terms  is  12.  What 
is  the  sum  of  all  the  terms  ? 

Explanation. — The  sum  of  the  extremes  of  an  arithmetical 
progression  being  equal  to  the  sum  of  any  two  terms  equally 
distant  from  them,  it  follows  that  the  terms  must  average  halt 
the  sum  of  the  extremes;  hence,  the  Sum  of  all  the  terms  equals 
the  product  of  the  number  of  terms  by  half  the  sum  of  the  extremes. 

2.  A  man  bought  25  acres  of  land:  for  the  first  acre 
he  gave  $i  ;  for  the  last,  $244^  ;  the  prices  of  the  suc- 
cessive acres  form  an  arithmetical  series.  How  much  did 
the  25  acres  cost  at  this  rate  ? 

3.  A  merchant  bought  25  pieces  of  cloth  ;  for  the  first 
piece  he  gave  $3  ;  for  the  last  piece,  $63  ;  the  prices  of 
the  pieces  form  an  arithmetical  progression.  How  much 
at  this  rate,  did  the  cloth  cost  him  ? 

4.  In  a  certain  field  there  are  44  rows  of  corn:  in  the 
first  row  there  are  10  hills  ;  and  in  the  last,  139  hills; 
the  number  of  hills  in  the  successive  rows  form  an  arith- 


224  PROGRESSION.  [CHAP.  IX. 

metical  progression.    How  many  hills  are  there  in  ^e 
field  ?  • 

•  CASE  III. 

Art.  225.  Given  the  extremes  and  the  common  dif- 
ference, to  find  the  number  of  terms. 

1.  The  first  term  of  an  arithmetical  progression  is  8; 
the  last  term  83  ;  and  the  common  difference  5.  What  is 
the  number  of  terms  ? 

Explanation. — Since  the  last  term  of  an  arithmetical  pro- 
gression equals,  the  product  of  the  number  of  terms  less  on© 
into  the  common  difference,  increased  by  the  first  term,  (see 
Case  1;)  it  follows  that  the  number  of  terms  equals  the  quotient^ 
increased  by  1,  arising  from  dividing  the  difference  of  the  extremes 
by  the  common  difference. 

2.  A  man  going  a  journey  traveled  the  first  day  1  miles, 
the  last  day  67  miles,  and  each  .day  increased  his  journey 
by  4  miles.     How  many  days  did  he  travel  ? 

3.  A  merchant  bought  a  certain  number  of  pieces  of 
cloth,  the  prices  of  which  increased  by  $2.  The  first 
piece  cost  $3,  and  the  last  piece  $43.  How  many  pieces 
did  he  buy  ? 

Geometrical  Progression. 

Art.  226.  A  series  of  numbers  that  increase  or  de- 
crease by  a  constant  fmiUiplier,  is  said  to  be  in  Geometrical 
Progression. 

When  the  constant  multiplier,  which  is  called  the  ratio, 
is  greater  than  a  unit,  the  series  is  called  an  Ascending 
Geometrical  Progression  ;  as, 

1,  2,  4,  8,  16,  32,  64,  128,  256,  512,  1024,  &c. 

When  the  ratio  is  less  than  a  unit,  the  series  is  called  a 
Descending  Geometrical  Progression  ;  as, 

512,  256,  128,  64,  32,  16,  8,  &c.;  or, 

-*■)  "a*  4>  Ji  T6»  ii2'»  "e*'  ^^' 
In  Geometrical,  as  in  Arithmetical  Progression,  there 
are  five  terms  to  be  considered  ; — 


ABT.  228.]  GEOMETRICAL  PROGRESSION.  225 

a,  The  first  term; 

1,    The  last  term; 

D,  The  number  of  terms; 

r,    The  ratio;  and 

s,   The  sum  of  all  the  terms. 

These  terms  are  so  related  that  any  three  of  them  being 
known,  the  remaining  two  may  be  found.  Since  there  are 
five  terms,  and  only  three  of  them  necessary  to  be  known, 
to  find  a  fourth;  it  follows  that  there  may  be  twenty  dis- 
tinct cases  in  Geometrical  Progression.  We  shall,  how- 
ever, notice  but  two  of  them,  and  refer  the  student  to 
Algebra  for  the  others. 

CASE    I. 

Art.  227.  Given  the  first  term,  the  number  of  terms, 
and  the  ratio,  to  find  the  last  term. 

1.  The  first  term  df'  a  geometrical  progression  is  2,  the 
ratio  is  3,  and  the  number  of  terms  is  8.  What  is  the 
last  term  ? 

Solution. — From  the  nature  of  geometrical  progression,  it 
is  evident,  that  the  second  term  equals  the  first  term,  multiplied 
by  the  ratio  ;  the  third  term  equals  the  first,  multiplied  by  the 
ratio  squared  ;  the  fourth  term  equals  the  firsts  multiplied  by 
the  ratio  cubed,  and  so  on  for  the  follo^ving  terms ;  Hence,  the 
first  term  multiplied  by  tliat  power  of  the  ratio  denoted  by  the 
number  of  terms,  less  one,  will  give  the  last  term. 

2.  A  person  traveling,  goes  3  miles  the  first  day,  6 
miles  the  second  day,  12  miles  the  third  day,  and  so  on, 
increasing  in  geometrical  progression,  for  6  days.  How 
far  did  he  go  the  last  day  ? 

3.  An  individual  commenced  business  with  a  capital  of 
$20,  and  was  so  fortunate  as  to  double  it  once  in  every 
two  years;  what  was  his  capital,  at  the  end  of  25  years? 

CASE    II. 

Art.  228.  Given  the  first  term,  the  last  term,  and  the 
ratio,  to  find  the  sum  of  all  the  terms. 

10* 


226  PROGRESSION.  [cHAP.  IX. 

1.  The  first  term  of  a  geometrical  progression  is  4,  the 
last  term  is  12500,  and  the  ratio  is  5.  What  is  the  sum 
of  all  the  terms  ? 

Explanation. — If  from  ratio  times  any  series,  as  2,  8,  and 
32,  we  subtract  the  scries,  the  remainder  will  be  ratio  times 
32 — 2.  It  is  also  evident,  that-  if  from  ratio  times  the  series, 
we  subtract,  once  the  series  there  will  remain,  (ratio — 1)  times 
the  series,  which  must  equal  ratio  times  32 — 2.  Therefore, 
the  sum  of  the  series  equals  4  times  32 — 2-i-(4 — 1.)  Hence, 
Multiply  the  last  term  by  the  ratio ;  from  the  product  subtract  the 
first  term  and  divide  the  remainder  by  the  rath  diminished  by  one, 
and  it  will  give  the  sum  of  all  the  terms. 

2.  A  gentleman  engaged  a  horse  and  carriage  to  ride 
LO  miles,  and  agreed  to  pay  ^  of  a  cent  for  the  first  mile;  1 
k?ent  for  the  second;  two  for  the  third  ;  and  so  on,  increas- 
ing in  geometrical  progression.  How  much  &t  this  rate, 
did  the  10  miles'  ride  cost  him  ? 

3.  A  speculator  sold  10  horses  o^this  condition  :  that 
he  should  pay  $3  for  the  first  horse;  $9  for  the-  second; 
$27  for  the  third,  and  so  on,  increasing  in  geometrical 
ratio.  What  did  the  last  horse  cost,  and  what  did  they 
all  cost  ? 

Summation  of  an  Infinite  Decreasing  Series.  An 
Infinite  Series  is  one  that,  being  continued,  would  run  on 
ad  infinitum.  If  a  decreasing  geometrical  series,  as  1,  ^, 
T'  i'  tV»  <^^-»  ^®  continued  to  infinity  the  last  term  evi- 
dently may  be  considered  0.  The  sum  of  such  a  series 
may  be  determined  as  follows  . 

Divide  the,  first  term  by  a  unit  diminished  by  the  ratio. 

1.  What  is  the  sum  of  the  infinite  series,  1,  i,  i,  -i,  &c.? 

2.  What  is  the  sum  of  the  infinite  series,  1,  i,  i,  &c.  ? 

3.  What  is  the  sura  of  the  infinite  series,  1,  |,  f,  &c.  ? 

4.  What  is  th^  sum  of  the  infinite  series,  1,  |,  -^\,  yf;^, 
&c.? 


^T. 


229.] 


INVOLUTION. 


227 


I 


CHAPTER  X. 
INVOLUTION  AND  EVOLUTION. 

Inyolution. 

Art.  229.  Involution,  teaches  the  method  of  raising 
a  number  to  any  proposed  power. 

The  number  to  be  raised  to  a  given  power  is  called  the 
Jirst  power,  or  root.  The  product  obtained  by  multiplying 
that  number  by  itself,  is  called  the  square,  or  second  jpower 
of  that  number. 


4  inches. 


Remark. — We  generally  say,  the 
square  of  a  number,  instead  of  the  second 
power  of  that  number,  because  the 
surface,  or  superficial  contents  of  a 
geometrical  square  is  obtained  by  mul- 
tiplying the  number  of  linear  units 
expressing  one  of  its  sides  by  itself. 
Thus,  the  side  of  the  adjacent  figure  is 
expressed  by  4  linear  units,  (4  inches) 
and  its  superficial  contents,  by  4x4=16 
square  indies. 

If  the  square  or  second  power  of  a  number  be  multiplied 
by  the  first  power  of  that  number,  the  product  is  called 
the  CUBE,  or  third  power  of  that  number. 

Remark. — We  generally  say,  the 
cube  of  a  number,  instead  of  the  third 
power  of  that  number,  because  the 
solid  contents  of  a  geometrical  cube  is 
expressed  by  the  third  power  of  the 
number  expressing  one  of  its  sides. 
Thus,  the  solidity  of  the  annexed  cube 
is  expressed  by  3  X  3  X  3  =  27  solid 
feet. 


The  power  to  which  a  number  is  to  be  involved  is  some- 
times expressed  by  a  small  figure,  called  an  exponent  or 


228  EVOLUTION,  [chap.  X. 

iiidex,  placed  above  and  a  little  to  the  right  of  that  num- 
ber.    Thus, 

52  =  5  X  5  =  25  the  square  of  5. 

53  =  5  X  5x  5  =  125  the  cube  of  5. 

54  =  6X5X5X5  =  625  the  fourth  power  of  5-. 

&c,,         &c.,         &c. 

The  exponent  of  a  quantity  shows  how  many  times  that 
uantity  mters  as  a  factor. 

Art,  230.  A  quantity  is  involved  to  any  given  power 
by  muliplying  it  by  itself  as  many  times  as  there  are  units  in 
the  expone7it,  less  one. 

1,  What  is  the  square  of  each  of  the  following  numbers  : 
1,  8,  9,  12,  25,  38,  274,  and  487  ? 

2,  What  is  the  cube  of  23,  84,  96,  and  273  respectively  ? 

3,  What  is  the  fourth  power  of  16,  18,  24, 147,  and  263, 
respectively  ? 

4,  What  is  the  fifth  power  of  3*2,  41-5,  82'5  and  829, 
respectively  ? 

5,  What  is  the  fifth  power  of  |,  |,  |  and  j,  respectively  ? 

6,  What  is  the  cube  of  3^,  2|,  4i,  and  14|,  respectively? 

Evolution, 

Art.  231.  Evolution  is  the  reverse  of  Involution. 
It  teaches  the  method  of  resolving  a  number  into  equal 
factors,  either  of  which  is  termed  a  root. 

The  square  root  of  49  ( =  7x7)  is  7,  since  7  is  one  of 
the  two  equal  factors  of  49. 

The  cube  root  of  27  (  =  3x3X3)  is  3,  since  3  is  one  of 
the  three  equal  factors  of  27, 

Numbers  whose  roots  can  only  be  approximately  ob- 
tained, are  called  surd  numbers. 

The  square  mot  is  indicated  by  the  symbol  ^,  and  is 
called  the  radical  sign.     Thus,  ^9  =  3  ;  ^25=5, 

The  cube  root  is  indicated  by  placing  the  figure  ^  above 
the  radical  sign.     Thus,  ^27=3  j  ^64=4. 


ART.  233.] 


SQUARE   ROOT. 


229 


SQUARE    ROOT. 

Art.  232.  The  square  root  of  any  number,  which  is  not 
a  surd,  may  be  determined  by 

Resolving  the  number  into  its  prime  factors — the  mntinued 
product  of  every  other  one  of  these  different  factors  judUI  he  thz 
root  required. 

1.  What  is  the  square  root  of  5184  ? 

OPERATION. 

5184  =  2x2*X2x2*x2x2*x3x3*x3x3* 


Explanation. — Every  other  one  of  the  different  prime  fac- 
tors of  5184  is  marked  by  *;  the  product  of  which  is  2  X  2  X 
2  X  3  X  3  =  72,  the  square  root  of  5184. 

2.  What  is  the  square  root  of  900  ? 

3.  What  is  the  square  root  of  18225  ? 

4.  What  is  the  square  root  of  396900  ? 

Art.  233.  The  square  root  of  any  quantity  which  is 
not  a  surd,  and  is  expressed  by  not  more  than  four  figures, 
can  be  ascertained  by  inspection. 

First,  square  the  nine  digits  respectively,  and  observe 
figure  of  each  square  number. 


the  terminating 


The  terminating  figures  that  are  alike 
are  linked  together. 

We  observe  that  all  square  numbers  end 
in  1,  4,  9,  6,  or  5;  also,  if  the  number  ends 
in  9,  the  figure  in  the  root  occupying  the 
unit's  place  must  be  either  3  or  7 ;  if  in  4, 
the  figure  in  the  root  occupying  the  unit's 
place  must  be  either  2  or  8,  &c.  The  fig- 
ures occupy  JDg  the  hundreds,  or  the  hunr 


230  SQUARE   ROOT.  [cHAP.  X 

dred's  and  thousa.id^s  place,  will  rnable  us  to  determine  the 
figure  of  the  root  occupying  the  ten's  place  ;  and  by  the 
excess  of  the  given  quantity  above  the  square  of  the  ten's 
figure,  we  are  enabled  to  tell  which  of  the  two  figures  that 
will  produce  the  terminating  figure  of  the  quantity,  is  the 
root. 

1.  What  is  the  square  root  of  5184  ? 

Remark — In  accordance  with  what  we  have  already  learned,  we  know  the 
figure  in  the*root  occupying  unit's  place  must  be  2  or  8  j  and  the  one  occu- 
pying the  ten's  place  must  be  7,  as  its  square,  49,  is  the  largest  square  number, 
which  is  less  than  61  ;  and  since  the  excess  of  the  51  above  49  is  so  small,  we 
take  tne  figure  2  for  the  unit's  figure  of  the  root.  Hence,  the  square  root  of 
the  above  number  is  72  Should  the  number  have  been  6084.  then  the  excess 
of  60  above  49  would  have  been  so  great,  we  should  have  taken  the  8  for  the 
unit" s  figure  of  the  root.  Hence,  we  would  have  73  for  the  square  root  of 
6084. 

2.  What  is  the  square  root  of  6t6  ? 

3.  What  is  the  square  root  of  2209  ? 

4.  What  is  the  square  root  of  1225  ? 

5.  What  is  the  square  root  of  2916  ? 

6.  What  is  the  square  root  of  3969  ? 
t.  What  is  the  square  root  of  5041  ? 

8.  What  is  the  square  root  of  1921  ? 

9.  What  is  the  square  root  of  8464  ? 
10.  What  is  the  square  root  of  9025? 

Art.  234.  The  square  of  1,  (the  smallest  digit,)  is  1. 
The  square  of  9,  (the  largest  digit,)  is  81.  Hence,  the 
square  of  any  digit  is  expressed  by  either  om  or  two 
figures. 

The  square  of  10  (the  smallest  number  denoted  by  two 
figures,)  is  100.  The  square  of  99,  (the  largest  number 
denoted  by  two  figures,)  is  9801.  Hence,  the  square  of 
any  number  denoted  by  two  figures,  is  expressed  by  either 
three  or  four  figures;  in  the  same  manner  it  may  be  shown, 
that  the  square  of  any  number  denoted  by  three  figures, 
will  be  expressed  by  either  Jive  or  six  figures,  &c. 

Hence,  the  square  of  any  number  will  contain  twice  as 
many  figures  as  that  number,  or  twice  as  m^ny,  less  one. 
Therefore,  to  extract  the  square  root,  we  first  separate  the  imrri' 
her  into  jperiods  of  tivo  figures  each,  commmcing  at  the  right 


ART.  236.] 


SQUARE  ROOT. 


231 


Art.  235.   As  Evolution  is  the  reverse  of  Involution, 

we  will  involve  a  few  quantities  by  considering  them  decom- 
posed into  UNITS,  TENS,  HUNDREDS,  &c.,  froui  wliich  we  will 
deduce  a  general  rule  for  the  extraction  of  the  square  root. 

The  square  of  a  binomial,  that  is,  a  quantity  consisting 
of  two  terms,  is  equal  to,  The,  square  of  the  first  term,  plus 
twice  the  first  term  iiito  the  second,  plus  the  square  of  the 
second  term. 

1.  What  is  the  square  of  35  ? 

35  =  30  +  5.     Consider  30  the  first_term  and  5  the 

second;  then  by  the  above  rule  we  have,  35'^=(30  +  5)  = 
30'+2x30x5  +  5'=:1225. 

The  evolution  by  multiplication  is  as  follows  : — 

30  4-5 
30  4-5 


3024-30  X  5 

30  X  5  +  5* 

30'4-2x30x5-f  5' 

This  involution  may  be  geometri-  E 
cally  illustrated  thus : — Suppose  the 
square  ABCD,  to  be  30  inches  each 
way;  then  its  superficial  contents 
is  expressed  by  30^  This  square 
may  be  increased  by  the  two  rec- 
tangles ABFE  andBCIH,  each  equal 
in  length  to  the  side  of  the  square, 
and  in  width  to  BH,  (or  5,)  the 
quantity  by  which  the  square  has 
been'increased ;  hence,  the  area  of 
each  of  these  rectangles  is  expressed  by  30  X  5  ;  also  the  little 
square  BHGF,  whose  side  is  BH,  (or  5)  ;  hence,  ^s  area  is  52. 

Art.  236.  The  square  of  any  polynominal  is  equal  to, 
The  square  cf  the  first  term,  plus  twice  the  first  term  into  the 
second,  plus  the  square  of  the  second  ;  plus  twice  the  sum  of 
the  first  two  into  the  third,  plus  the,  square  of  the  third  ;  and 
so  on. 


232 


SQUARE   ROOT. 


[chap.  X. 


(4004-50)  X2 

?1 

400x50 

50^ 

X 

o 

^ — \ 

\o 

^  ! 

400' 

X 

o 

+ 

o 

o 

■^ 

^ 

1.  What  is  the  square  of  452  ? 

452  =  400  +  50  +  2.  Consider  400  the  first  term, 
50  the  second  term,  and  2  the  third  term;  then  by  the 
above  theorem  we  have,  (400  +  50  +  2'i2=:400'+2x400 
X50+50'+2X(40Cf50)X2  +  22. 


The  following  diagram  exhibits 
the  above  involution  geometri- 
cally. 


By  reversing  the  above  process  of  involution,  we  obtain 
for  exTfacting  the  square  root,  the  following 

GENERAL    RULE. 

Commencing  at  the  right,  separate  the  number  into  periods 
of  two  numbers  each. 

Find  the  greatest  square  number  in  the  first  period  on  the 
left,  and  place  its  root  at  the  right  of  the  number,  in  the  form 
of  a  quotient ;  also,  on  the  left  separating  it  from  the  num- 
ber by  a  perpendicular  line.  Then  subtract  the  square  of 
this  root  from  the  period  on  the  left ;  and  to  the  remainder 
annex  the  second  period  ;  which  will  form  the  first  dividend. 

Double  the  root  already  found,  (which  is  placed  at  the 
left  of  the  number) ;  to  this  product  annex  a  cipher,  and  it 
will  form  the  first  trial  divisor.  The  number  of  times  the 
trial  divisor  is  contained  m  the  first  dividend^  will  be  the 
next  figure  of  the  root,  which  must  be  added  to  the  trial 
divisor,  to  form  the  true  divisor.  Multiply  the  true  divisor 
by  the  figure  of  the  root  last  obtained  ;  subtract  the  product 
from  the  dividend,  and  to  the  remainder  annex  the  n£xt  period 
for  a  NEW  dividend. 

To  the  last  divisor,  add  the  last  figure  of  t/ie  root  found , 


ART.  236.] 


SQUARE  ROOT. 


233 


this  sum  with  a  djpher  annexed  will  be  the  next  trial  divisor. 
Then  proceed  as  before,  until  all  the  periods  have  been  brought 

down. 

Note. — When  any  dividend  is  not  so  large  as  its  trial  divisor,  place  a  cipher 
for  the  next  figure  of  the  root ;  also,  place  a  cipher  at  the  right  of  the  divisor, 
and  form  a  new  dividend  by  annexing  a  new  period. 

1.  What  must  be  the  length  of  the  side  of  a  square 
pond  that  shall  contain  54756  square  feet  ? 

OPERATION. 
Number.  Root. 

Linear  ft.  Sq.  ft.  Linear  ft. 


Ist  trial  divisor, 


200 
400 


true  divisor,    430 
2d  trial  divisor,    460 

true  divisor,    464 


54756(200  +  30  +  4  =  234. 
40000 


14756 
12900 

1856 
1856 


N 


K 


H 


M 


Explanation The    re-        A  J    I 

quirement  of  the  above  ques- 
tion was  to  determine  the 
side  of  a  square  that  should 
contain  54756  square  feet. 

It  is  evident  that  the  side 
of  the  square  must  be  more 
than  200  linear  feet,  since 
the  square  of  200  is  less  than 
54756  :  also,  that  it  must  be 
less  than  300  linear  feet, 
since  the  square  of  300  is 
greater  than  54756.  There-  D 
fore,  2  is  the  greatest  num- 
ber whose  square  is  contain- 
ed in  5,  (the  left  hand  period,)  and  is  the  first,  or  hundreds' 
figure  of  the  root. 

Let  CDEL  be  a  square  whose  side  is  200  linear  feet.  Then 
its  area  is  200'^  =  40000  square  feet,  which  being  taken  from 
the  given  number,  leaves  14756  square  feet,  to  be  added  to  the 
square  DL.     We  first  add  the  two  rectangles  CN  and  EM 


200 


E      F  G 


234  SQUARE  ROOT.  [cHAP.  X. 

each,  equal  in  length  to  the  side  of  the  square  DL,  which  has 
already  been  found  to  be  200  feet.  Therefore,  the  length  of 
the  two  rectangles  is  400  feet,  which  forms  the  1st  trial  divisor. 
The  14750  being  divided  by  the  1st  trial  divisor,  gives  a  quo- 
tient of  30,  which  is  the  width  of  the  rectangles  CN  and  EM, 
also,  the  length  of  the  side  of  the  small  square  LMKN.  Add- 
ing to  400,  (the  length  of  the  two  rectangles  CN  and  EM,)  30, 
(the  length  or  side  of  the  square  LK,)  gives  430,  the  true 
divisor.  Multiply  430,  the  length  of  these  three  pieces  by  30, 
their  width,  gives  12900  square  feet,  the  amount  by  which  the 
square  DL,  has  been  increased.  Subtract  this  amount  from 
14756  square  feet  leaves  1856  square  feet,  to  be  added  to  the 
square  BDFK. 

We  now  add  to  the  square  DK,  the  two  rectangles  BJ  and 
FH,  each  equal  in  length  to  200,  (the  side  of  the  square  DL.) 
plus  30,  (the  width  of  the  rectangles  just  added  to  the  square 
DL).  Therefore,  2  (200  +  30)  ==  460  linear  feet,  is  the  length 
of  the  two  rectangles  BJ  and  FH,  which  forms  the  2d  trial 
divisor.  Divide  1856,  (the  number  of  square  feet  remaining 
to  be  added  to  the  square  DK.)  by  the  2d  trial  divisor,  gives  4 
for  the  width  of  the  two  rectangles ;  also  the  side  of  the  small 
square  KHIJ.  Hence,  the  length  of  the  two  rectangles  BJ  and 
FH,  increased  by  the  length  of  the  square  KHIJ,  is  464,  which 
forms  the  last  true  divisor  ;  this  length  being  multiplied  by  4, 
the  width  of  the  three  pieces,  gives  1856  square  feet,  which 
being  taken  from  1856  leaves  no  remainder.  Therefore,  the 
square  ADGl,  the  side  of  which  is  200  -f-  30  -f-  4  =  234  linear 
feet,  contains  54756  square  feet. 

By  omitting  the  ciphers  the  foregoing  operation  will 
take  the  following  condensed  form  : — 

OPERATION. 
Number.       Root. 
Linear  ft.       Sq.  ft.     Linear  ft. 


2 

43 

464 


54756(234 
4 

147 
129 

1856 
1856 


ART.    238.]  '  SQUARE    ROOT.  23jGt 

2.  What  is  the  square  root  of  85264  ? 

3.  What  is  the  square  root  of  55696  ? 

4.  What  is  the  square  root  of  1971216  ? 

5.  What  is  the  square  root  of  5499025  ? 

6.  What  is  the  square  root  of  269222464  ? 
1.  What  is  the  square  root  of  6497004816  ? 

8.  What  is  the  square  root  of  609596100  ? 

9.  What  is  the  square  root  of  4164081009664  ? 

Remark. — The  roots  of  the  above  numbers  can  also  be  determined  by  Art 
232. 

Art.  237.  To  extract  the  square  root  of  a  decimal. 

Commencing  at  the  decimal  paint,  separate  the  number  into 
periods  of  two  figures  each;  then  proceed  according  to  the 
General  Rule^  and  to  the  number  annex  ciphers^  until  the 
desired  number  of  figures  in  the  root  are  obtained. 

1.  What  is  the  square  root  of  223024  ? 

2.  What  is  the  square  root  of  64-1601  ? 

3.  What  is  the  square  root  of  84-65901  ? 

4.  What  is  the  square  root  of  187'20924  ? 

5.  What  is  the  square  root  of  5296  ? 

6.  What  is  the  square  root  of  2  ? 

7.  What  is  the  square  root  of  3  ? 

8.  What  is  the  square  root  of  5  ? 

9.  What  is  the  square  root  of  6  ? 

10.  What  is  the  square  root  of  7  ? 

11.  What  is  the  square  root  of  9  ? 

Atr.  238.  To  extract  the  square  root  of  a  common 
fraction. 

Reduce  the  fraction  to  its  simplest  form;  then  extract 
the  root  of  the  numerator  and  denominator  separately,  if 
they  have  an  exact  root;  if  not,  reduce  the  fraction  to  a 
decimal,  and  proceed  as  in  Art,  237. 

1.  What  is  the  square  root  of  -//t  ^ 

2.  What  is  the  square  root  of  ||J|  ? 

3.  What  is  the  square  root  of  f  |  ? 

4.  What  is  the  square  root  of  |  ? 

5.  What  is  the  square  root  of  |  ? 


236  SQUARE  ROOT.  [CHAP.  X. 

QUESTIONS  INVOLVING  THE  PRINCIPLES  OP  SQUARE  ROOT. 

Art.  239.  A  triangle  is  a 
figure  having  three  sides,  and 
therefore  three  angles.  When 
one  of  the  angles  is  right,  like  the 
corner  of  a  square,  (that  is,  con- 
tains 90°,)  the  triangle  is  called  ease. 
a  right-angled  triangle.     The 

side  opposite  the  right-angle,  is  called  the  hypothenuse  ;  one 
of  the  remaining  two  sides  is  called  the  last,  and  the  other 
the  perpendicular. 

Art.  240.  Two  sides  of  a  right-angled  triangle  being 
given,  the  third  side  can  be  found  by  means  of  the  follow- 
ing theorem. 

It  is  an  established  theorem  of  geometry.,  that  the  square  of  the 
hypothenuse  is  eqimlto  the  sum.  of  the  squares  of  the  other  two  sides. 

Tlierefore,  the  square  of  one  of  the  sides  is  equal  to  tlie  square  of 
the  hypothenuse,  diminished  by  the  square  of  the  other  side. 

1.  How  long  must  a  ladder  be  to  reach  to  the  top  of  a 
tree  52  feet  high  when  the  foot  of  it  is  39  feet  from  the 
tree? 

OPERATION. 

62'  =  2704 
39'  =  1521 

^^4225  =  65  feet,  the  length  of  the  ladder. 

2.  It  is  ascertained  that  a  ladder  95  feet  in  length, 
standing  on  the  bank  of  a  river  57  feet  in  width,  reaches 
to  the  top  of  a  tree  standing  on  the  opposite  bank.  What 
is  the  height  of  the  tree  ? 

3.  Two  ships  start  from  the  same  place  and  sail,  the  one 
North  and  the  other  East.  How  far  apart  will  they  be 
in  6  days,  providing  they  sail,  at  the  rate  of  72  and  96 
miles  an  hour,  respectively  ? 

4.  A  man  standing  39  paces,  of  3  feet  each,  from  a  tree 
which  is  95  feet  high  and  6  feet  in  diameter,  shoots  a 
pigeon  from  its  topj  how  far  did  the  ball  move  before  it 


ART.  241.] 


PRACTICAL    QUESTIONS. 


23t 


reached  the  pigeon,  providing  the  man's  eye,  the  place 
from  which  the  ball  started,  is  five  feet  above  the  ground  ? 
5.  What  is  the  distance  between  the  opposite  corners 
of  a  parallelopipedon,  the  length  of  which  is  8  feet,  and 
the  width  and  depth,  each  6  feet  ? 

Remark. — The  question  will  be  more  readily  comprehended  by  inspecting 
the  following  diagram 

From  the  right-angled  triangle  IHE       ^  jj 

we  determine  the  hypothenuse,  IE. 
Then,  from  the  right-angle  triangle 
lEB,  we  determine  the  hypothenuse, 
IB,  which  is  the  distance  betwee^  the 
opposite  corners  of  the  parallelopipe- 
don, DE. 

Art.  241.  The  three  smallest  integers  that  can  accu- 
rately express  the  length  of  the  sides  of  a  right-angled 
triangle,  are  3,  4,  and  5. 


\^    / 

\^ 

/ 

s 

/^\! 

Thus- 


A 


If  we  multiply  these  three  numbers  by  2,  it  will  give  a 
right-angled  triangle,  the  sides  of' which  are  6,  8,  and  10; 
if  by  3,  another,  the  sides  of  which  are  9,  12,  and  15.  In 
the  same  way,  any  number  of  triangles  may  be  obtained, 
the  sides  of  which  are  expressed  by  integers. 

Hence,  by  knowing  two  sides  of  a  right-angled  triangle, 
the  sides  of  which  are  to  each  other  as  3,  4,  and  5  we  can 
readily  determine  the  remaining  side,  mentally. 

1.  What  must  be  the  length  of  a  ladder  to  reach  to  the 
top  of  a  tree,  48  feet  high,  when  its  foot  is  placed  36' feet 
from  its  base  ? 

2.  What  is  the  distance,  between  the  opposite  corners 
of  a  rectangular  field,  the  length  of  which  is  32  rods,  and 
the  width  24  rods  ? 

3.  What  is  the  length  of  a  rectangular  field,  the  dis- 
tance between  the  opposite  corners  of  which  is  30  rods, 
and  the  width  of  which  is  18  rods  ? 


238 


SQUARE    ROOT. 


[chap.  X. 


Mechanical  Application  of  the  Foregoing. 

Art.  242.  Mechanics  generally  make  use  of  a  right- 
angled  triangle,  the  sides  of  which  are  6,  8,  and  10  feet, 
respectively,  in  squaring  the  walls  for  the  foundation  of  a 
building,  &c.  This  is  done  by  placing  an  upright  stick 
where  we  design  the  corner  of  the  building  to  be,  with  a 
cord  about  it  so  as  to  form  a  plain  angle ;  then  measure 
off  6  feet  on  one  end  of  the  cord,  and  8  feet  on  the  other, 
and  holding  the  cord  horizontal,  place  the  terminating 
point  of  the  6  feet  (which  may  be  Tuarked  by  stickii  g  a 
pin  through  the  cord,)  at  one  extremity  of  a  ten-foot 
pole,  and  the  terminating  point  of  the  8  feet  at  the  other 
extremity.  The  triangle  thus  formed  will  be  a  right- 
angled  triangle. 

The  following  diagram  will  render 
the  above  remark  more  plain.  P  re- 
presents the  upright  stick,  about 
which  the  cord  is  placed.  PA,  the 
6  feet  measured  off,  PB,  the  8  feet, 
and  AB,  the  ten-foot  pole. 


Length  of  Braces. 


AB  is  the  corner  post  of  a  build- 
ing; DG  a  girth  ;  and  CE  a  brace. 
The  triangle  CDE  is  a  right-angled 
triangle  :  hence,  the  length  of  tire 
brace  CE  is  found  by  extracting  the 
square  root  of  the  sum  of  the  squares 
of  the  two  lengths  DE  and  DC.  (See 
Art.  240.) 


Art.  243.  The  length  of  any  brace,  v.ihen  DC  and 
DE  are  of  the  same  length,  is  equal  to  the  length  DC 
-\-  as  many  times  5  inches  as  DC  is  feet  in  length.  This 
fact  is  of  great  practical  utility  to  carpenters. 

1.  What  is  the  length  of  a  brace,  when  the  two  sides 
DC  and  DE  are  each  3  feet  long  ? 


ART. 


24.4.] 


LENGTH    OF    BRACES. 


239 


The  length  of  the  brace  will  be  3  feet  +  3  times  5 
inches,  equal  to  3  feet  +  15  inches  =  4  feet  3  inches. 

2.  If  the  sides  DC  and  DE  are  each  4  feet  long,  3^  feet 
long,  4i  feet  long,  5  feet  long,  5^  feet  long,  or  6  feet  long, 
what  would  be  the  length  of  tlie  braces  for  the  several 
conditions,  respectively. 


Length  of  Rafters,  &c. 

Art  244.  To  find  the  length 
of  a  rafter. 

FB  is  called  the  base  line  of  the 
roof,  and  HM  the  height  of  the 
pitch  of  the  roof. 

If  the  height  of  the  pitch  is  equal 
to  om-half  of  the  base  line;  HM=: 
MB;  hence,  the  length  of  the  raf- 
ter, HB,  is  found  in  the  same  man- 
ner we  found  the  length  of  a  brace, 
under  Art.  243. 

A  roof  is  said  to  be  one-fon,rth,  two-fifths,  three-sevenths^ 
&,c., pitch,  when  HM  =  ^,  f ,  -f ,  &c.,  of  the  base  line,  FB. 
The  length  of  the  rafters  of  such  roofs,  is  found  by  extracting 
the  square  root  of  the  sum  of  the  squares  of  JIM  and  MB. 
(See  Art.  240.) 

1.  In  a  three-eighths  pitch  roof,  the  base  of  which  is  40 
feet,  what  is  the  length  of  the  rafters  ? 

-In  this  example  the  height  of  the  pitch  HM= 


Solution. 
15  feet. 


_    The  MB  =  20  feet. 
152  =  225  feet. 
202  =  400  feet. 

The  square  root  of  625  =  25,  the  length  of  the  rafters. 

2.  In  a  two-seventh  pitch  roof,  the  length  of  the  base 
of  which  is  28  feet,  what  is  the  length  of  the  rafters  ? 

3.  In  a  one-fourth  pitch  roof,  the  base  of  which  is  24 
feet,  what  is  the  length  of  the  rafters  ? 


240  SQUARE   ROOT.  [CHAP.    X. 

4.  In  a  three-fifths  pitch  roof,  the  base  of  which  is  40 
feet,  what  is  the  lenorth  of  the  rafters  ? 

Art.  245.  It  is  an  established  theorem  of  geometry,  t/iat 
all  similar  surfaces,  or  areas,  are  to  each  other  as  the  squares 
of  their  like  dimensions. 

Hence,  the  like  dimensions  of  similar  figures  are  to  each 
othtt  as  the  square  roots  of  their  areas. 

1.  There  are  two  circular  fish-ponds  ;  one  of  which  is  20 
rods  in  diameter,  and  the  other  4  rods  in  diameter.  How 
much  more  surface  in  the  one  than  in  the  other  ? 

2.  A  farmer  has  a  rectangular  piece  of  land  containing 
61  acres,  the  width  of  which  is  10  rods,  and  the  length 
100  rods.  His  neighbor  has  a  similar  piece  of  land  con- 
taining  9  acres.  Required  the  length  and  breadth  of  hi8 
neighbor's  piece  of  land. 

3.  Suppose  a  horse  to  be  tied  to  a  post  in  the  centre  of 
a  field,  by  a  rope  7'13  rods  in  length,  and  is  thereby  ena- 
bled to  graze  upon  1  acre.  How  long  should  the  rope  be 
to  allow  it  to  graze  upon  6^  acres  ? 

4.  By  observation  I  find  that  11-i-  gallons  of  water  will 
flow  through  an  orifice  of  1^  inches  in  diameter  in  1  second. 
How  large  should  the  orifice  be  so  as  to  discharge  2J-  gal- 
lons in  the  same  time. 

5.  If  it  require  156^  yards  of  carpet  to  cover  a  floor 
that  is  25  feet  in  length,  and  20f  feet  in  width  ;  what 
must  be  the  dimensions  of  a  similarly  shaped  floor,  that 
requires  56|-  yards  of  the  same  kind  of  carpet  to  cover  ? 

6.  Five  men  purchased  a  grindstone  40  inches  in  dia- 
meter. How  much  of  the  diameter  must  each  grind  off, 
so  as  to  have  \  of  the  stone  ? 

Remark. — After  the  first  has  ground  off  his  share,  |  of  the 
Btone  remains,  and  its  diameter  will  be  40^i  =  8^20,  &c. 

Art.  246.  When  the  base  and  the  sum  of  the  height 
and  hypothenuse  of  a  right-angled  triangle  are  given,  to  find 
the  hypothenuse  : 

Add  the  square  of  the  height  and  hypothenuse  to  the  square 
of  the  base,  and  divide  their  sum  by  twice  tJie  height  and 


4RT.    241] 


CUBE   ROOT. 


241 


1.  There  is  a  tree  80  feet  in  height,  standing  by  the 
bank  of  a  river  50  feet  in  width.  Where  must  this  tree 
break  off,  so  that  the  top  will  reach  across  the  river,  while 
the  broken  parts  remain  in  contact  ? 


CUBE  ROOT. 

Art.  247.  Whenever  the  cube  root  of  a  quantity  is 
expressed  by  a  whole  number,  it  may  be  found  by, 

Resolving  the  number  into  its  prime  factors.  The  product 
of  every  third  factor  of  these  different  factors,  will  he  the 
root  required. 

1.  What  is  the  cube  root  of  129000  ? 

OPERATION. 

2)729000 


( 


Explanation. — Taking  the  continued  product 
of  every  third  one  of  these  different  factors, 
(which  are  marked  by  *  )  we  have  2x3x3 
X  5  ==  90,  which  is  the  cube  root  of  729000. 


2)364500 

^2) 182250 

3)91125' 

3)30375 

»3) 10125 

3)3375 

3)1125 

*3)375 

5)125 

5)25 


*5 

2  What  is  the  cube  root  of  4741632  ? 

3  What  is  the  cube  root  of  98611128  ? 
4.  What  is  the  cube  root  of  621875  ? 

11 


242  CUBE  ROOT.  [chap.  X. 

5.  What  is  the  cube  root  of  2388t8t2  ? 

6.  What  is  the  cube  root  of  5639752  ? 

1.  What  is  the  cube  root  of  5936493568  ? 

Art.  248.  The  cube  root  of  any  quantity  which  is 
not  a  surd  and  is  expressed  by  not  more  than  six  figures, 
can  be  ascertained  by  inspection. 

First,  cube  the  nine  digits  respectively,  and  observe  tho 
terminating  figure  of  each  cube  number. 

Digits.       Their 
Cubes. 

13  =s       1       It  will  be  observed  that  the  terminating  figure 

2'  =       8  of  each  of  the  cubes  of  the  nine  digits  is  either 

33  =    2  7  1,  2;,  3,  4,  5,  6,  7,  8,  or  9 ;  hence,  every  cube  num- 

43  =    6  4  ber  must  terminate  with  one  of  the  nine  digits, 

53  =  12  5  consequently  the  figure  of  the  root  occupying  the 

63  =  21  6  unit's  place  is  readily  determined  by  inspection. 

73  =  34  3  The  figure  of  the  root  occupying  the  tens  place 

8^  =  51  2  is  determined   by  inspecting  the  number,  con- 

93  =  72  9  sidered  as  units,  that  preceed  the  first  three  figures. 

1.  What  is  the  cube  root  of  614125  ? 

ExPLANATio,N. — As  this  number  ends  in  5  the  figure  in  the 
root  occupying  the  unit's  place  must  be  5.  8  is  the  largest 
number,  the  cube  of  which  is  less  than  the  number  expressed 
by  the  figures  on  the  left  of  the  first  three  figures,  which  ia 
614;  hence,  the  cube  root  of  614125  is  85. 

2.  What  is  the  cube  root  of  8593t  ? 

3.  What  is  the  cube  root  of  226981  ?     • 

4.  What  is  the  cube  root  of  117649  ? 
6.  What  is  the  cube  root  of  50653  ? 
6.  What  is  the  cube  root  of  110592  ? 
1.  What  is  the  cube  root  of  405224  ? 

8.  What  is  the  cube  root  of  438976  ? 

9.  What  is  the  cube  root  of  778688  ? 

Art.  249.  Before  we  attempt  to  explain  the  usual 
method  of  extracting  the  cube  root,  we  will  involve  a 
number,  consisting  of  units  and  ^erw,  and  of  units,  tmsf 
and  hundreds  to  its  third  power. 


ART.    260.]  '  CUBE    llOOT.  243 

Remark.— The  cube  of  I,  the  smallest  digit  is  1.  The  cube  of  9,  the  largest 
digit  is  7-29.  Therefore,  the  cube  of  any  digit  is  expressed  by  one.  two,  or  tinet 
figures. 

The  cube  of  10.  the  smallest  number  denoted  by  two  figures,  is  1000.  The 
cube  of  99,  the  largest  number  denoted  by  Iwo  figures,  is  970299.  Therefore, 
the  cube  of  any  number  denoted  by  two  figures  is  expres.sed  by  fouk,  five, 
or  SIX  tigtires.  In  the  same  maniier  it  may  be  shown,  that  the  cube  of  a  num- 
ber denoted  by  three  figures  is  expressed  by  sevk.n,  Eir.H  r,  or  ninf.,  figures,  &c. 

Hence,  if  a  number  be  denoted  by  one.  two,  or  th-ee  figures,  its  cube  root  will 
be  expressed  by  one  figure  ;  if  hy  four,  Jive,  or  six  figures,  its  cube  root  will 
be  expressed  by  two  figures,  &c. 

In  general  the  cube  will  contain  three  times  as  many  figures  as  theroot,  or  three 
times  as  many  less  one  or  two.  Therefore,  to  extiuct  the  cube  root,  we  first 
separate  tlie  number  into  periods  of  three  figures  each,  commencing  at  the  right. 

Art.  250.  The  cube  of  a  Binomial,  (that  is,  a  num- 
ber consisting  of  two  terms,)  is,  the  cube  of  the  first,  or 
left  hand  term,  plus  three  times  the  square  of  the  first  term 
into  the  second,  plus  three  times  the  first  term  into  the  square 
of  the  second  term,  plus  the  cube  of  the  second  term. 

In  general,  the  cube  of  any  Polynomial  is  equal  to  the 
cube  of  the  first,  or  left-hand  term,  plus  three  times  the  square 
of  the  first  term  into  the  second,  plus  three  times  the  first  into 
the  square  of  the  second,  plus  the  cube  of  the  second ;  plus 
three  times  the  square  of  the  sum  of  the  first  two  into  the 
third,  plus  three  times  the  sum  of  tJie  first  two  into  the  square 
of  the  third,  plus  the  cube  of  the  third,  Sfc. 

1.  What  is  the  cube  of  89  ? 

89  =  80  +  9;  ami  by  Art._250,  we  have 

(80  +  9)  3  =  80'  +  3  X  80'  X  9  +  3  X  80  X  92  +  9» 

2.  What  is  the  cube  of  3ot  ? 
357  =  300  +  5.0  +  7j_therefore; 

(300  +_50  +  1)3=  3~00'+  3  X  300'  X  50  +  3  X  300X 
60'  +  50'  X  3(300  +  50)2  ^  7  +3(30(^  +  50)  X  1*^  +  1' 

3.  What  is  the  cube  of  468  ? 

468=400  +  60  +  8;.-., _      _ 

(400  +  60+8)3=400'+3x400'x60  +  3X400x60'+60^ 
+  3(400  +  60)^x8+3(400  +  60)  X  82+83 

The  involution  of  example  1,  by  multipJ»caMoa.  is  as 
follows: — (the  second,  &c.,  is  similar  to  it). 


244 


CUBE    ROOT. 


[chap.  X 


80  +  9 
80  +  9 


80'+80X9 


80x9  +  95 


(80  +  9)2=80'+2X  80x9  +  92 
80+9 


80^+2X80x9+80X92 


80  X9  +  2X80X92+93 


(80  +  9)3=80'+3x80'x  9+3x80x9^+93 

We  will  now  illustrate  geometrically  the  involution  of  the 
first  example. 

How  many  cubic  feet  in  a  cube,  each  side  of  which  is 
89  feet  ? 

89=80  +  9. 

Fig.  1.  D 


Suppose  each  side  of  the  cube  AD, 
(fig.  1,)  to  be  80  feet;  then  its  solid 

contents  will  be  80^  =  512000  cubic 
feet. 


To  increase  the  size  of  the  cube 
AI),  we  will  first  add  the  three 
square  slabs,  AC,  BD,  and  CE, 
each  of  the  sides  of  which  is  80 
feet,  (the  side  of  the  cube  AD,) 
and  the  thickness  of  each,  9  feet. 
Hence,  the  solid  contents  of  one 
of  these  square  slabs  is  80^^  X  9, 
and  the  three,  3  X  80^^  X  9  = 
172800  cubic  feet. 


ABT.  260.] 


CUBE   ROOT. 


245 


Fig.  2. 


The  cube  AD,  increased  by  the 
three  square  slabs,  AC,  BD,  and 
CE,  is  represented  by  Fig.  2 ; — 
the  contents  of  which  is  612000 
4-172800  =  684800  cubic  feet. 


We  will  now  increase  Fig.  2, 
by  the  three  equal  corner-pieces, 
AB,  BC,  and  CD,  the  length  oi 
each  being  80  feet,  (the  side  of 
the  cube  AD,)  and  the  width  and 
thickness  of  each,  9  feet.  Hence, 
the  solid  contents  of  one  of  these 
pieces  is  80  X  92,  and  of  the 
three,  3  X  80  X  9^  =  19440  cubic 
feet 


Fig.  2,  increased  by  the  three 
pieces,  AB,  BC,  and  CD,  is  rep- 
resented by  Fig.  3  ; — the  contents 
of  which  is  512000  -f  172800  X 
19440  =  704240  cubic  feet. 


We  will  now  increase  Fig.  3, 
by  the  small  cube  XY,  each  side 
of  which  is  9  feet ;  therefore,  its 
contents  is  9^  =  729  cubic  feet ; 


This  cube  is  then  represented  by 
Figure  4,  the  contents  of  which  is 
512000  -f-  172800  -f  19440  -f 
729  =  704969  cubic  ""eet,  and 
the  side  of  which  is  8t  feet. 


1   '•  / 


246  CUBE  ROOT.  [chap.  X. 

By  reversing  the  above  process,  we  obtain  for  extracting 
the  cube  root,  the  following 


GENERAL   RULE. 

CommeTicing  at  units,  separate  the  number  into  periods  of 
three  figures  each. 

Then  find  the  largest  digit,  the  cube  of  which  shall  not 
exceed  the  left-hand  period.  Place  this  digit,  which  is  called 
the  first  figure  of  the  root,  on  the  right,  in  the  form  of  a 
quotient ;  also,  on  the  left,  for  the  first  term  of  a  first  column, 
and  its  square  for  tM  first  term  of  a  second  column,  and  from 
the  left-hand  period  of  the  given  number,  subtract  its  cube. 
Then  to  the  rcTnainder,  annex  the  mxt  period,  for  the  first 
DIVIDEND.  JYow  double  the  term  in  the  first  column,  for  its 
second  term,  and  add  its  product  into  the  root  already  found, 
to  the  first  term  of  the  second  column,  for  the  first  trial 
DIVISOR.  Consider  two  ciphers  annexed  to  the  trial  divisor, 
and  write  the  number  of  times  it  is  contained  in  the  divi- 
dend, for  the  next  figure  of  the  root ;  also,  annex  it  to 
Hue  sum  of  the  last  term  in  the  first  column,  and  the  first 
figure  of  the  root ; — this  will  be  the  next  term  of  the  first 
column.  Add  the  product  of  this  term  into  the  digit  of  the 
root  last  found,  advancing  it  two  places  to  the  right,  to  the 
last  term  of  the  second  column,  for  its  next  term  ;  this  will  be 
the  TRUE  DIVISOR.  From  the  dividend,  subtract  the  product 
of  the  true  divisor  into  the  digit  of  the  root  found ;  and  to 
the  remainder  annex  the  nzxt  period,  for  the  second  dividend. 

Proceed  in  a  similar  way  until  all  the  periods  have  been 
used. 

Remark. — By  carefully  examining  the  foregoing  involution,  the  pupil  will 
be  able  to  deduce  other  rules  for  the  extraction  of  the  cube  root,  some  of 
which  may  perhaps,  appear  more  plain  than  the  one  I  have  just  given,  as  this 
is  more  readily  deduced  from  Algebraic  involutions.  I  have  given  this  rule, 
as  it  will  be  less  laborious  to  extract  the  cube  root  of  large  numbers  by  it, 
than  by  many  other  rules  usually  given  ;  also,  because  it  keeps  distinct  the 
three  geometrical  magnitudes — lines,  surf acec,  and  solids. 

'J'he  Jirst  rule,  however,  is  the  most  simpk;,  and  will  ^e  found  of  much  im- 

1)0  tance  in  reducing  surd  Quantities  to  their  simplest  form,  (as  will  hereafter 
)f  fifxplained,)  or  in  determining  the  roots  of  rational  quantities. 

1.  What  is  the  cube  root  of  104969  ? 


RT.  260.] 

CUBE  ROOT. 
OPERATION. 

2 

First  Col. 

Sfcond  Col. 

NuMBKR.              Root. 

Unear  ftet. 

S<iuare  feet. 

Cubic  feet.        Linear  feet. 

80 
160 

6400 
19200,  trial  divisor. 

704969(80  -f  9  =  89 
512000 

249 

21441,  true  divisor. 

192969,  1st  dividend. 
192969 

24T 


0 

Explanation. — We  first  find  the  greatest  cube  contained  in 
the  left-hand  period.  We  know  that  this  number  must  be 
more  than  80,  since  80^  =  512000,  which  is  less  than  704969; 
also,  that  it  must  be  less  than  900,  since  90^  =  729000,  which 
is  greater  than  704969.  Hence,  the  first,  or  left-hand  figures 
of  the  root,  is  8  ;  whose  cube  is  512,  which  is  the  greatest  cdbe 
contained  in  704,  the  first  or  left-hand  period. 

Fig.  1.  D 


Let  each  side  of  the  cube  AD, 
represented  by  Fig.  1,  be  80 
linear  feet ;  then  its  cubical  con- 
tents will  be  80^  =  512000  cubic 
ft. ,  and  704969—512000=192969 
cubic  feet,  which  is  still  to  be 
added  to  the  cube  AD. 


We  first  add  the  three  square 
slab  pieces  AB,  BC,  and  CD, 
whose  length  an"d  breadth  are 
each  respectively  80  feet,  (the 
side  of  the  cube  AD.)  The  area 
of  the  face  of  the  first  piece,  AB, 
is  80^  =  6400  square  feet.  The 
length  of  the  other  two  pieces, 
BC,  and  CD,  is  80  -f-  80  =  160 
feet,  and  their  width  80  feet. 
Hence,  their  superficial  contents 
is  160  X  80  =  12800  square  feet, 
which  added  to  6400  square  feet, 
the  superficial  contents  of  the 
piece,  AB,   gives  19200    square 


248 


CUBE   ROOT. 


[chap. 


feet,  the  superficial  contents  of 
the  three  pieces,  AB,  BC,  and 
CD.  As  these  three  square  slabs 
make  up  by  far  the  greatest 
amount  of  the  whole  increase,  if 
we  divide  192969,  (the  number 
of  cubic  feet  remaining  to  be 
added,)  by  192000,  the  number 
of  square  feet  in  the  three  pieces, 
AB,  BC,  and  CD,  (which  may  be 
called  the  trial  divisor,)  it  will 
give  their  thickness ;  which  we 
find  to  be  9  feet. 

Figure  2,  represents  the  cube 
AD,  with  the  three  pieces  AB, 
BC,  and  CD,  added. 

We  now  add  the  three  corner- 
pieces,  E(j,  HF,  and  HX,  whose 
lengths  are  respectively  80  feet, 
(the  side  of  the  cube  AD,)  and* 
whose  width  and  thickness  are 
each  9  feet  respectively ;  also, 
the  corner-piece  AW,  whose 
length,  width,  and  thickness,  are 
each  9  feet.  Therefore,  the  length 
of  the  three  pieces,  EG,  HF,  and 
HX,  is  240  feet ;  which  being  in- 
creased by  the  length  of  Ihe  cor- 
ner-piece, AW,  gives  249  feet  for 
the  length  of  the  four  pieces. 
Their  width  is  9  feet ;  therefore, 
249  X  9  =  2241  square  feet,  is 
their  superficial  contents ;  which 
being  increased  by  19200  square 
feet,  the  superficial  contents  of 
the  three  pieces  already  added 
on,  gives  21441  square  feet,  (the 
superficial  contents  of  the  seven 
pieces  added  on,)  which  being 
multiplied  by  9,  their  thickness, 
gives  192969  cubic  feet,  their 
solid  contents,  which  being  sub- 
tracted from  192969  cubic  feet, 
the  quantity  that  remained  to  be 
added  to  the  cube  AD,  leaves  no 
remainder ;  therefore,  a  cube 
represented  by  Figure  3,  whose 
side  is  80  -f  9  =  89  feet,  will 
cantain  704969  cubic  feet. 


Fig.  3. 


^=- 



7^ 

^ 

^ 

ijl|iil!l:JI!:liiii 

|:iillli!lll:';llii 

!l!lll! 

lii-' 

l.:i! 

ART.  251.]  CUBE   ROOT.                                         249 

This  work  may  be  condensed  by  omitting  the  ciphers  and 
unimportant  terms. 

First  Col.  Second  Col.                              Niimhei:  Root. 

8  64           704969(89 

16  192           512 


249         21441  192«69 

192969 


2.  What  is  the  cube  root  of  12895213625  ? 

Remark.— To  render  the  method  of  extracting  the  cube  root  familiar,  when 
there  are  a  number  of  figures  in  the  root,  we  will  perform  the  above  example 

First  Col.  Second  Col.  Number.  Root. 

2  4  12895213625(2345 

8 

4  12  

63       1389        4895 
66       1587        4167 


694      161476       728213 
698       164268       645904 


7025      16461925      82309625 

82309625 


Remark. — When  the  trial  divisor  is  greater  than  its  corresponding  dividend, 
place  0  for  the  next  figure  of  the  root,  and  bring  down  the  next  period.  Then 
use  the  same  trial  divisor  with  two  more  ciphers  annexed. 

3.  What  is  the  cube  root  of  46964099891t  ? 

Remark. — Whenever  the  number  has  not  an  exact  root,  there  will  be  a  re- 
mainder after  the  last  period  has  been  brought  down.  The  process  may  b* 
continued,  and  the  true  root  more  nearly  obtained,  by  annexing  ciphers  to 
new  periods.     The  figures  thus  obtained  will  be  decimals. 

4.  What  is  the  cube  root  of  1860867  ? 

5.  What  is  the  cube  root  of  469640998917  ? 

6.  What  is  the  cube  root  of  58050510848  ? 

7.  What  is  the  cube  root  of  84672374  ? 

8.  What  is  the  cube  root  of  3  ? 

9.  What  is  the  cube  root  of  5  ? 

10.  What  is  the  cube  root  of  7  ? 

11.  What  is  the  cube  root  of  493780134751068073294S  « 

Art.  251.  To  extract  the  cube  root  of  a  decimal. 
First,  commmcing  at  the  decimal  'point,  separate  the  numbers 

11* 


250  CUBE    ROOT.  [chap.    X. 

into  periods  of  three  figures  each  ;  if  necessary,  annex  cipher s^ 
so  that  tibe  decimal  may  he  separated  into  equal  periods.  Thmi 
-proceed  as  usual. 

12.  What  is  the  cube  root  of  561-515625  ? 

13.  What  is  the  cube  root  of  460-099648  ? 

14.  What  is  the  cube  root  of  -1U649  ? 

15.  What  is  the  cube  root  of  t-256313856  ? 

16.  What  is  the  cube  root  of  41-86  ? 

Art.  252.  To  extract  the  cube  root  of  a  fraction,  or 
mixed  number,  first  reduce  either  of  them  to  its  simplest 
form;  then  find  the  root  of  the  numerator  and  denominator 
separately,  if  their  roots  can  be  accurately  found  ;  if'  not, 
reduce  the  fraction  to  a  decimal,  and  extract  the  root  as 
above  directed. 

It.  What  is  the  cube  root  of  -||||  ? 

18.  What  is  the  cube  root  of  ifffffff-  ? 

19.  What  is  the  cube  root  of  ^\^  ? 

Art.  253.  A  rational  quantity  is  one  that  can  be  ex- 
pressed in  numbers.     Thus,  ^"11  is  rational. 

An  irrational  or  surd  quantity  is  one  that  has  not  an 
exact  root,  or  which  cannot  be  expressed  in  numbers. 
Thus,  ^,3/3  is  a  surd. 

When  the  cube  root  of  a  large  quantity  is  required,  it 
will  be  found  more  convenient  to  find  the  root  of  the 
rational  part  of  the  number  by  Art.  247,  and  the  root  of 
the  surd  part  by  General  Rule.     - 

Thus,  ^40=^2X2 x2x"5  =  2^5  =  2xl'7099764-  = 
3'419952  +  . 

1.  What  is  the  cube  root  of  19208000  ? 

OPERATION. 

2)19208000 


2)9604000 
*2)48020'00 


[over. 


ART.  253.]  PRACTICAL    QUESTIONS.  251 

2)2401000 
2)1200500 
^2)600250 


5)300125 


7)2401 


Remark. — Every  third  factor  is  marked  thus,  *. 
Their  product  will  be  the  cube  root  of  the  ra- 
5)60025     tional  part,  which  is  2x2x5x7  =  140.     This 
^5)  12005     multiplied  by  ^/TTwill  equal  the  cube  root  of  the 
above  number.    Thus,  1404/7=140  X 19 12931-f- 
=267-81034-1-. 
7)343 

*7)49 

2.  What  is  the  cubfe  root  of  128625  ? 

3.  What  is  the  cube  root  of  61631955000  ? 

4.  What  is  the  cube  root  of  257250  ? 

5.  What  is  the  cube  root  of  84035  ? 

6.  What  is  the  cube  root  of  2401000  ? 

PRACTICAL    QUESTIONS    IN    CUBE    ROOT. 

Art,  253.  It  is  a  theorem  of  geometry,  that  all  similar 
folids  are  to  each  other  as  the  cubes  of  their  like  dimensions. 
Therefore, 

The  dimensions  of  similar  solids  are  to  each  otJier  as  thz 
CUBE  ROOTS  of  their  solidity. 

1.  If  a  ball  of  iron  2  inches  in  diameter  weigh  5  pounds, 
what  will  a  ball  of  the  same  metal  weigh,  the  diameter  of 
which  is  7  inches. 

OPERATION. 

2^     :      73    :  :     5  :  weight  required. 
8     :    343  :  :     5  :  214^  pounds. 

2.  If  the  Earth  is  8000  miles  in  diameter,  and  Mercury 
3200  miles,  how  many  times  larger  is  the  Earth  than 
Mercury  ? 


252  GAUGING.  fCHAP.    X. 

3.  If  a  ball  |  of  an  inch  in  diameter  weigh  ^  of  a  pound, 
what  must  be  the  diameter  of  another  ball  of  the  same 
metal  to  weigh  24  pounds  ? 

4.  What  is  the  diameter  of  a  globe  of  gold  that  is 
worth  $9T85036"80,  providing  a  pound  of  gold  (avoirdu- 
pois weight,)  is  worth  $256  ? 

It  will  be  remembered  that  the  specific  quantity  of  gold  is   1S| ;  and  that  a 
cubic  foot  of  water  weighs  62^  pounds. 

6.  If  a  man  5^  feet  in  height,  weigh  125  pounds,  how 
tall  is  that  man  who  weighs  216  pounds  ? 

6.  A  cask  60  inches  long  and  36  inches  at  the  bung 
diameter,  contains  a  certain  number  of  gallons  ;  what 
must  be  the  dimensions  of  a  similar  cask  that  shall  contain 
\  as  much  ? 

1.  A  carpenter  wishes  to  make  a  cubical  cistern  tl^t 
shall  contain  14088  cubic  feet  of  water;  what  must  be  the 
length  of  one  of  its  sides  ? 

8.  If  a  cellar  12  feet  long,  8  feet  deep  and  8  feet  wide 
contain  768  cubic  feet ;  what  is  the  dimensions  of  a  similar 
cellar  that  shall  contain  20736  cubic  feet? 

9.  What  will  be  the  dimensions  of  a  rectangular  box, 
which  shall  contain  1845480  cubic  feet,  the  length, 
breadth,  and  depth  being  to  eaeh^ther  as  7,  5,  and  3  ? 

10.  A  ball  of  fine  thread  5  inches  in  diameter  is  owned 
by  five  women  ;  what  portion  of  the  diameter  must  each 
wind  off  so  as  to  share  equally  of  the  thread  ? 

Gauging. 

Art.  254.  Gauging  teaches  the  method  of  finding  the 
contents  of  any  regular  vessel,  in  gallons,  bushels,  &c. 

Art.  255.  To  find  the  number  of  gallons  or  busliels  in 
a  square  vessel. 

Take  the  dimensicms  in  inches,  and  divide  the  product 
arising  from  mvltiplying  the  length,  breadth,  and  height 
together  by  2S2  for  ale  gallons,  231  fyr  wine  gallons,  and 
21 50-42 /or  hushels. 


ART.    257.]  GAUGING.  253 

1.  How  many  wine  gallons  will  a  cubical  box  contain, 
that  is  5  feet  long,  2^  feet  wide  and  2  feet  high  ? 

2. How  many  ale  gallons  will  a  vess  el  contain,  that  is 
9  feet  long,  4  feet  wide,  and  3  feet  high  ? 

3.  How  many  bushel  of  grain  will  a  bin  contain,  that 
is  16  feet  long,  12  feet  wide,  and  6  feet  high  ? 

Art.  256.  To  find  the  contents  of  casks. 

Multiply  the  'product  of  the,  square  of  the  mean  diaTmter 
and  the  length  in  inches,  hy  '0034,  and  it  will  give  the  con- 
tents in  wine  gallons  ;  if  by  "0028  instead  of  *0034,  it  will 
give  the  contents  in  beer  gallons. 

Remark. — The  mean  diameter  is  equal  to  the  head  diameter  incrsased  hy 

-i.  of  the  difference  between  the  head  and  bunec  diameters  when  the  slaves 

1 TT 

are  m%ich  curved,  or  by  adding  1  the  difference,  when  but  little  curved  ;  and 


1.  How  many  wine  gallons  does  a  cask  contain  whose 
length  is  34  inches,  bung  diameter  28  inches,  and  head  dia- 
meter 20  inches  ? 

2.  How  many  wine  gallons  does  a  cask  contain,  whose 
length  is  45  inches,  bung  diameter  32  inches  and  its  head 
diameter  22  inches  1 

Art.  257.  To  find  the  contents  of  a  round  vessel, 
wider  at  one  end  than  the  other. 

To  i  of  the  square  of  tM  difference  of  tht  diameters,  add 
their  product  and  multiply  this  sum  hy  the  height.  Then 
multiply  hy  "0034  for  wine  gallons,  and  hy  '0028  for  ale 
or  beer. 

1.  How  many  wine  gallons  will  a  vessel  contain,  that  is 
48  inches  in  diameter  at  the  bottom  and  32  inches  at  the 
top  ;  the  length  of  which  is  60  inches. 

2.  How  many  beer  gallons  will  a  tub  contain,  that  is 
40  inches  in  diameter  at  the  bottom  and  30  inches  at  the 
top  ;  the  length  of  which  is  48  inches. 


254 


MENSURATION. 


[chap.    XL 


CHAPTER  XI. 


MENSURATION. 

Geometrical  Definations. 

1.  A  Point  is  that  which  has  position,  but  not  magnitude. 

2.  A  Line  is  length  without  width  or  thickness,  as  AB. 

a ^B 

3.  Parallel  Lines  are  those  which  are  everywhere 
equally  distant,  as  AB  and  BO.  ^ » 


4.  A  Surface  is  that  which  has  r 
length  and  breadth  without  thickness,  [ 
as  ABCD.  B 


5.  A  Solid  Body  is  that  which  has  length, 
breadth,  and  thickness  :  and  therefore  com- 


bines 
AB. 


the  three  dimensions  of  extension,  as 


Plane  Figures. 

Art.  258.  1.  A  Plane  Figure  is  a  plane  surface  ter- 
minated on  all  sides  by  lines,  either  straight  or  curved. 

2.  A  Polygon,  or  rectilineal  figure,  is  a  phme  terminated 
on  all  sides  by  straight  lines.  The  sum  of  these  bounding 
lines  is  called  the  contour  or  jperimeter  of  the  polygon. 

3.  A  Triangle  is  a  b 
figure  having  three  sides 
and  three  angles,  as  ABC. 
Its  altitude  is  a  line  let 
fall  from  the  vertex  per- 
pendicular to  the  base, 
asBD. 


ART.  258.]  GEOMETRICAL   DEFINITIOXS. 


4.  A  Square  is  a  figure  that  has  all  of  its 
Bides  equal,  and  its  angles  right-angles,  as 
CDEG. 

The  line  CE  is  called  its  diagonal. 


5.  A  Parallelogram  is  a  figure  that 
has  its  opposite  sides  parallel,  as  ABCF. 


6.   A  Rectangle  is  an   equiangular 
parallelogram,  as  CYDG. 


7.  A  Trapezoid  is  a  figure 
that  has  only  two  of  its  sides 
parallel,  as  XYBZ. 


Remark. — It  has  alreadj-  been  remarked,  that  any  figure,  the  sides  of  which 
are  terminated  by  stiaigfit  lines,  is  called  a  Polygon. 

A  polygon  of  three  sides  is  called  a  Triangle  ;  that  o{  four  sides  a  Quadri- 
lateral ;  that  of  ^fe  sides,  a  Pentagon;  that  of  six,  a  Hexagon  ;  that  of  seven, 
a  Heptagon  ;  that  of  eight,  an  Octagon  ;  that  of  nine,  a  Nonagon  ;  that  of  ten, 
a  Decagon  ;  that  of  twelve,  a  Dodecagon,  &c. 


A  Circle  is  a  plane,  termi- 
nated by  a  curved  line,  every 
point  of  which  is  equally  distant 
from  a  point  within,  called  the 
centre.  The  curved  line  is  called 
the  circumference. 


The  Diameter  of  a  circle  is  a  line  passing  through  the 
centre,  and  terminated  by  the  circumference,  as  AC. 

The  radius  of  a  circle,  is  a  line  drawn  from  its  centre  to 
the  circumference,  as  BD,  BE,  &c.;  hence,  it  is  half  the 
diameter. 


256 


MENSURATION. 


[chap.   XI. 


9.  An  Ellipse  is  a  plane 
bounded  by  a  curved  line,  the 
sum  of  the  distances  from  eve- 
ry point  of  which  to  two  given 
points,  is  equal  to  a  given  line. 
The  line  AB,  is  called  the 
transverse,  and  CD,  the  covr 
jugate  axis. 

Solid  Figures. 

Art.  259.  A  Prism  is  a 
solid,  the  sides  of  which  are 
parallelograms,  and  the  ends 
equal  and  parallel  polygons, 
as  figure  A. 

Remarks.— When  the  ends  of  a  Prism  are  triangular,  it  is  called  a  triangular 
•prism;  when  they  are  squares,  it  is  called  a  square  prism,  &c. 

B 

2.  A  Cube  is  a  j)rism,  all 
the  sides  of  which  are  equal 
squares,  as  figure  B. 


3.  A  Parallelopipedon  is  a 


prism 


the  ends  of  which  are 


parallelograms,  as  figure  C. 

4.  A  Cylinder  is  a  solid, 
having  equal  circles  for  ends, 
and  is  generated  by  the  revo- 
lution of  a  rectangle  about  one 
of  its  sides;  as  figure  D 


J 


5.  A  Pyramid  is  a  solid,  having  for  its  base 
a  plane  rectilinear  figure ;  and  for  its  sides  tri- 
angles, whose  vertices-  meet  in  a  point  at  the 
top,  called  the  vertex  of  the  pyramid.  Figure 
E  represents  a  triangular  pyramid. 


ART. 


261.] 


MENSURATION   OF   SURFACES. 


25t 


6.  A  Cone  is  a  solid,  having  for  its  base  a 
circle,  and  tapers  uniformly  to  a  point  at  the  top. 
Figure  F  represents  a  cone. 


*l.  A  Frustum  of  a  pyramid y 
or  a  cone,  is  the  part  that  remains 
after  cutting  off  the  top  by  a 
plane  parallel  of  the  base. 

Fig.  M  represents  the  frus- 
tum of  a  pyramid;  and  N,  that 
of  a  cone. 


8.  A  Sphere  is  a  solid,  bounded  by  a 
convex  surface,  every  point  of  which  being 
equally  distant  from  a  point  within  called 
a  center.     Figure  X  represents  a  sphere. 


Mensuration  of  Surfaces,  &c. 

Art.  260.  The  pupil  is  referred  to  Geometry  for  the 
demonstration  of  the  following  rules  for  measuring  surfaces, 
solids,  &c. 

Art.  261.  The  area  of  a  figure  is  the  number  of  square 
inches,  feet,  or  yards,  &c.,  which  it  contains. 

Problem  1 Given  the  base  and  altitude  of  a  triangle,  to 

find  its  area. 

Multiply  the  base  by  half  the  altitude. 

1.  What  is  the  area  of  a  triangle  whose  base  is  9  feet, 
and  altitude  4  feet  ? 

2.  What  is  the  area  of  a  triangle  whose  base  is  36  rods, 
and  altitude  16  rods  ? 


25B 


MENSURATION. 


[chap.    XI. 


Problem  2. — Given  the  three  sides  of  a  triangle,  to  find  its 


From  half  of  the  sum  of  the  three  sides,  subtract  each  side 
separately  ;  mid  the  square  root  of  the  continued  product  of 
these  three  remainders  aiid  the.  halj%sum  will  be  the  area. 

1.  What  is  the  area  of  a  triangle  whose  sides  are,  re- 
spectively, 12,  18,  and  20  feet? 

2.  What  is  the  area  of  a  triangular  field  whose  sides  are 
respectively  40,  30,  and  50  rods  ? 

Problem  3 Given  the  sides  of  a  rectangle,  or  square,   to 

find  its  area. 

Multiply  the  lerigth  by  the  width. 

1.  How  many  square  feet  of  boards  will  be  required  to 
floor  a  room  that  is  36  feet  long  and  18  feet  wide  ? 

2.  How  many  acres  in  a  rectangular  piece  of  land  460 
rods  long  and  380  wide  ? 

3.  How  many  more  acres  in  a  piece  of  land  160  rods 
square,  than  in  a  rectangular  piece  40  rods  in  length  an^ 
32  rods  in  width. 

Problem  4. — Given  the  base  and  altitude  of  a  parallelogram, 
to  find  its  area. 

Multiply  the  base  by  the  altitude. 

1.  What  is  the  area  of  a  parallelogram  whose  length  is 
28  feet,  and  altitude  22  feet  ? 

Problem  5.— Given  the  altitude  and  the  parallel  bases  of 
a  trapezoid,  to  find  its  area. 

Multiply  the  altitude  by  half  the  'mm  of  its  parallel  sides. 

1.  What  is  the  area  of  a  trapezoidal  field,  the  parallel 
sides  of  which  are  18  and  24  rods  respectively; — the  per- 
pendicular distance  between  these  sides  being  8  rods  } 

Rkmark.— This  rule  is  of  practical  use  to  lumbermen  in  measuring  boards. 

Let  ABCD  be  a  tapering 
board,  and  EG  its  length. 
The  half  sum  of  its  parallel 
sides,  AB  and  CD,  is  HK,  the 
width  of  the  board  nt  the 
middle  point.  Its  area,  there- 
fore.is  expressed  by  HKXE^O. 


ART.  261.] 


MENSURATION  OF  SURFACES. 


259 


2.  How  many  square  feet  in  a  tapering  board  36  feet 
long,  18  inches  wide  at  one  end  and  32  in.  at  the  other  ? 

Problem  6. — Given  the  diameter  of  a  circle  to  find  its  cir- 
cumference. 

Miiltijply  the  diameter  by  3'1416. 

1,  What  is  the  circumference  of  a  circle  whose  diameter 
is  15  feet  ? 

2.  What  is  the  circumference  of  a  circle  whose  diameter 
is  24  rods  ? 

Remark. — The  true  ratio  of  the  circumference  of  a  circle  to  its  diameter 
has  never  yet  been  found,  hs  approximate  value  has  been  extended  to  more 
than  200  places.  A  man  by  the  name  of  Van  Ceukn  first  extended  the  approxi- 
mation to  36  places  by  means  of  continually  bisectinpf  the  arc  of  a  circle. 
This  was  considered  so  great  an  achievement  that  the  36  numbers  expressing 
the  circumference  of  a  circle  whose  diameter  is  1,  was  engraved  on  his 
tomb-stone. 

The  following  are  the  numbers  : — 
3-141592653589793238462643383279502884 

Problem  7, — Given  the  circumference  of  a  circle  to  find  its 
diameter. 

Divide  the  circumference  by  8*1416. 

1.  What  is  the  diameter  of  a  circle  whose  circumference 
is  62-832  feet  ? 

2.  What  is  the  diameter  of  a  circle  whose  circumfer 
ence  is  48'5  feet  ? 

Problem  8. — Given  the  diameter  of  a  circle,  to  find  its  area. 
Multiply  the  square  of  the  diameter  by  -7854. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  15 
inches  ? 

2.  What  is  the  area  of  a  circle  whose  diameter  is  36  ft.  ? 


Remark. — From  the  above  rule,  we  can 
readily  determine  the  area  contained  between 
two  concentric  circumferences. 

It  is  also  worthy  of  remark  that  the  area  of 
a  .square  is  to  the  area  of  an  inscribed  circle, 
as  1  is  to  0.7854. 


260  MENSURATION.  [CHAP.  XI. 

Problem  9.— Given  the  diameter  of  a  circle,  to  find  the  side 
fan  equal  square. 

Muliijffly  the  diameter  Jy  .8862. 

1.  A  gentleman  has  a  circular  fish-pond  that  is  8  rods 
in  diameter  ;  what  must  be  the  side  of  a  square  pond  that 
shall  contain  the  same  area  ? 

2.  I  have  a  circular  piece  of  board  that  is  36  inches  in 
diameter  ;  what  must  be  the  side  of  a  square  board  that 
shall  contain  the  same  area  ? 

Problem  10. — Given  the  diameter  of  a  circle  to  find  the  side 
of  an  inscribed  square. 

Multiply  the  diarnet^  by  0.^0*11, 


1.  Required  the  side  of  a  square 
that  can  be  inscribed  in  the  circle 
ABO,  whose  diameter  is  24  feet. 


2.  How  large  a  square  stick  can  be  sawn  from  a  piece 
of  round  timber,  that  is  42  inches  in  diameter  ? 

Problem  11. — Given  the  diameters  of  an  ellipse,  to  find  its 
area. 

Multiply  the  product  of  its  two  axes  by  0'*I854. 

1.  How  many  square  rods  in  an  elliptical  garden,  whos(, 
transverse  axis  is  280  feet,  and  conjugate  axis  210  feet  ? 

2.  How  many  square  feet  in  an  elliptical  table,  the 
transverse  axis  of  which  is  6  feet  9  inches,  and  the  con- 
jugate axis  3  feet  6  inches  ? 

Problem  12. — Given  the  length  and  the  circumference  of  a 
cylinder,  to  find  its  surface. 

Multiply  its  circumference  hy  its  length,  a  :\d  to  the  product 
add  the  area  of  the  two  bases. 


ART.  261.]  MENSURATION    OF    SURFACES.  '      261 

1.  What  is  the  surface  of  a  cylinder  whose  T^ength  is  22 
feet,  and  diameter  4^  feet  ? 

Problem  13. — Given  the  length  and  the  perimeter  of  the 
base  of  a  prism,  or  parallelopipedon,  to  find  its  surface. 

Multiply  its  length  by  the  perimeter  of  its  base,  arid  to  the 
product  add  the  areas  of  the  two  ends. 

1.  What  is  the  surface  of  a  square  prism  whose  side  is 
2  foot  8  inches,  and  length  16  feet  ? 

2.  What  is  the  surface  of  a  triangular  prism  whose 
length  is  12  feet,  and  whose  sides  are,  respectively,  3,  4, 
and  5  feet  ? 

Problem  14. — Giyen  the  side  of  a  cube  to  find  the  area  of 
its  surface. 

Multiply  the  area  of  one  of  its  sides  by  6. 

1.  What  is  the  area  of  a  cubic  block,  the  side  of  which 
is  12  feet  ? 

Problem  15. — Given  the  slant  height  and  the  sides  of  the 
base  of  a  pyramid  to  find  its  area^ 

To  the  area  of  the  triangles  that  form  its  sides,  add  the 
area  of  the  base. 

1.  What  is  the  area  of  a  triangular'^pyramid,  the  slant 
height  of  which  is  12  feet,  and  the  sides  of  its  base  3,  4 
and  5  feet,  respectively  ? 

'  2.  What  is  the  area  of  a  square  pyramid,  the  slant 
height  of  which  is  35  feet,  and  the  sides  of  its  base  10 
feet? 

Problem  16 Given  the  slant  height  and  the  diameter  of  a 

cone,  to  find  its  surface. 

Multiply  the  half  sum  of  the  slant  height  and  the  radius 
of  the  base  by  the  circumference  of  the  base. 

1.  What  is  the  surface  of  a  cone,  the  slant  height  of 
which  is  47  feet  and  the  diameter  18  feet  ? 

Problem  17. — Given  the  perimeter  of  the  bases  and  tho 


262  MENSURATION.  [cHAP.   XI. 

slant  height  of  the  frustum  of  a  pyramid,  or  cone,  to  find  the 
area. 

Multiply  the  half  sum  of  the  perimeter  of  the  bases  by  the 
slant  height,  arid  to  the  product  add  the  sum  of  the  areas  of 
the  two  bases. 

1.  Suppose  the  slant  height  of  a  square  frustum  is  24 
feet,  the  side  of  the  base  8  feet,  aud  of  the  upper  base  or 
top  4  feet;  what  is  its  surface  ? 

2.  Suppose  the  slant  height  of  the  frustum  of  a  cone 
to  be  40  feet,  the  diameter  of  the  base  15  feet,  and  that 
of  the  top  5  feet  ;  what  is  its  whole  surface  ? 

Problem  18. — Given  the  diameter  of  a  sphere  to  find  its 
area  or  convex  surface. 

Multiply  its  circumference  by  its  diameter.  Or,  which  is 
the  same  thing, 

Multiply  the  square  of  the  diameter  by  3'1416. 

1.  What  is  the  surface  of  a  sphere  48  inches  in  diame- 
ter ? 

2.  What  is  the  area  of  the  earth's  surface,  supposing  it 
to  be  8000  miles  in  diameter  ? 

Art,  262.  We  can  readily  determine  the  distance  to 
any  visible  object  by  means  of  a  right-angled  triangle. 

Suppose  a  man  standing  at  A,  de- 
siring to  know  the  distance  to  any 
object  as  B,  on  the  opposite  side  of 
a  river  ;  how  should  he  proceed  to 
determine  this  distance,  providing  he 
has  nothing  but  a  ten-foot  pole  ? 

Form  the  right-angle  BAG,  and 
measure  any  distance,  as  AC  =  30 
feet.  Then  from  C,  measure  towards 
A  any  distance,  as  CD  =3  feet  ;  also 
DE,  perpendicular  to  CA.  Suppose 
DE=:4  feet.  Then  by  similar  triangles 
we  have 


ART,  262.]  MENSURATION    OF    SOLIDS.  263 

CD  :  CA  :  :  DE   :  AB.     Or, 

3     :  30     :  :  4       :  40,  the  distance   AB. 

The  above  principle  is  also  employed 
in  measuring  the  heights  of  trees,  &c. 
In  this  case  let  AC  be  the  height  of  the 
tree  ;  E  the  height  of  the  man's  eye, 
which  we  will  suppose  =  5  feet ;  DG  a 
perpendicular  pole,  and  DH  the  height 
above  the  eye  Then  by  similar  tri- 
angles we  have 

EH  :  EB  ::  HD  :  BC.     Suppose  we  e 
have  found  the  first  three  terms  of  this 
proportion  to  be  as  follows  : — 

3  :  24  ::  4  :  CB.  Then  CB  =  32  ft.  which  being  increased 
by  AB  =  5  ft.  we  have  32  +  5=37  ft.  the  height  of  the  tree. 

Mensuration  of  Solids. 

Problem  19. — Given  the  side  of  a  cube,  to  find  its  solHity. 

Multijply  its  lengthy  hreadth,  and  depth  together. 

1.  What  is  the  solidity  of  a  cube,  the  side  of  which  ir  S  ft.? 

Problem  20. — Given  the  length,  and  dimensions  of  t^^  end 
of  a  prism,  or  parallelopipedon,  to  find  its  solidity. 

Multiply  the  area  of  its  base  or  end  by  its  length. 

1.  What  is  the  solidity  of  a  triangular  prism  24  ^*. 
long,  and  each  side,  1  i  feet  ? 

2.  What  is  the  solidity  of  a  stick  of  timber  36  feet  lor,'' 
and  2|  feet  square  ? 

Problem  21. — Given  the  altitude  and  the  base  of  a  pyramid 
to  find  its  solidity. 

Multiply  the  area  of  its  base  by  otie  third  of  its  altitude. 

1.  What  is  the  solidity  of  a  triangular  pyramid,  the  al- 
titude of  which  is  24  feet,  and  each  side  of  the  base  6  ft.  ? 


264  MENSURATION.  [CHAP.    XI 

2.  What  is  the  solidity  of  a  square  pyramid,  the  altitude 
of  which  is  48  feet,  and  a  side  of  the  base  9  feet  ? 

Problem  22. — Given  the  altitude  and  the  diameter  of  a  cone, 
to  find  its  solidity. 

Multiply  the  area  of  its  base  hy  one-third  of  its  .altitude. 

1.  What  is  the  solidity  of  a  cone,  the  altitude  of  which 
is  16  feet,  and  the  diameter  of  the  base  3  feet  ? 

2.  What  is  the  solidity  of  a  cone,  the  altitude  of  which 
is  54  feet,  and  the  diameter  of  the  base  12  feet  ? 

Problem  23. — Given  the  altitude,  and  the  dimensions  of  the 
two  bases  of  a  frustum  of  a  pyramid,  or  of  a  cone,  to  find  its 
solidity. 

To  the  sum  of  the  areas  of  the  two  bases,  add  the  mean 
proportional  betv^een  these  two  areas,  and  multiply  the  result 
by  one-third  of  the  altitude  of  the  frustum. 

Remark. — A  mean  proportional  between  the  areas  of  the  two  bases  of  a 
cone,  is  the  square  root  of  the  product  of  the  squares  of  the  two  diameters, 
into  6-7854. 

1.  What  is  the  solidity  of  the  frustum  of  a  cone,  whose 
altitude  is  15  feet,  and  whose  bases  are  10  and  5  feet  in 
diameter,  respectively  ? 

OPBRATION. 

102 X 0-7854=100  X 0-7854,  the  area  of  the  lower  base. 

52X0-7854=  25x0*7854,  the  area  of  the  upper  base. 


-v/lOOX 25X0- 7854=  50x0-7854,    ^area  of  the  mean   proportional 
\         between  the  bases. 

175x0-7854   t^®  sum  of  the  area  of  the  three 


Hence, 

175  X  0-7854 X^  of  15=687*225,  cubic  feet,  the  solidity  of  the  cone. 

Remark. — This  rule  is  of  much  importance  in  determining  the  solidity  of 
round  sticks  of  timber,  the  diajneter  of  the  ends  of  which  dilJer. 

2.  There  is  a  stick  of  timber,  in  the  form  of  the  frustum 
of  a  cone,  that  is  48  feet  long,  4  feet  in  diameter  at  the 
larger  end,  and  9  inches  at  the  smaller  end.  How  many 
cubic  feet  does  it  contain  ? 

Art.  26^.  Another  method  of  measuring  round  tim- 
ber, Is  tb 


ART.   263.]  MENSURATION    OF    SOLIDS.  265 

Multiply  the  square  of  oTie-fourtk  the  girth,  in  inches,  hy 
the  length  of  the  stick,  in  feet  ;  then  divide  the  product  by  144. 
The  quotient  will  be  the  contents  in  cubic  feet. 

Remark.— The  girth  should  be  taken  two-thirds  of  the  distance  from  the 
smaller  to  the  larger  end. 

3.  How  many  cubic  feet  in  a  stick  of  timber,  25  feet 
long,  and  whose  girth  is  60  inches  ? 

4.  How  many  cubic  feet  in  a  stick  of  timber  42  feet 
ong,  and  whose  girth  is  80  inches  ? 

6.  How  many  cubic  feet  in  a  stick  of  timber  36  feet  long, 
and  whose  girth  is  48  inches  ? 

Problejm  24. — Given  the  diameter  of  a  sphere,  to  find  its 
solidity. 

Multiply  its  surface  by  one-sixth  of  its  diameter.     Or, 
Multiply  the  cube  of  the  diameter  by  0  5236. 

1.  What  is  the  solidity  of  the  Earth,  supposing  its  dia- 
meter to  be  8000  miles  ? 

2.  How  many  cubic  inches  in  a  cannon  ball  9  inches  in 
diameter  ? 

It  is  believed  that  the  foregoing  rules  will  enable  the 
pupil  to  solve  most  of  the  examples  that  may  arise  in 
ordinary  mensuration. 

For  the  practical  convenience  of  those  who  have  occasion 
to  refer  to  mensuration,  we  subjoin  the  following 

Table  of  Multiples  for  Mechanics. 

1.  Diameter  of  a  circle  X  31416  =  Circumference. 

2.  Radius  of  a  circle  X  6-2S3185  =:  Circumference. 

3.  Square  of  the  radius  of  a  circle  X  3-1416  =  Area. 

4.  Square  of  the  diameter  of  a  circle  X  0  7854  =  Area. 

5.  Circumference  of  a  circle  X  0159155  =:  Radius. 

6.  Square  root  of  the  area  of  a  circle  X  0  50419  =  Radius. 

7.  Circumference  of  a  circle  X  0-31831  =  Diameter. 

8.  Square  root  of  the  area  of  a  circle  X  1  •12338  =  Diameter. 

9.  Radius  of  a  circle  X  1732051  =  Side  of  inscribed  eqiiilateral  triangle. 

10.  Diameter  of  a  circle  X  0860254  =  Side  of  inscribed  equilateral  triangle. 

11.  Side  of  inscribed  equilateral  triangle  X  0  577350  =  Radius  of  circle. 

12.  Radius  of  a  circle  X  1-414214  =  Side  of  inscriDed  square. 

13.  Diameter  of  a  circle  X  0-7071  =  Side  of  an  inscribed  square. 


266 


MENSUEATIOX. 


CHAP.    XI 


14.  Side  of  inscribed  square  X  0  707107  =  Radius. 

15.  Square  of  radius  of  a  sphere  X  12  566371  =  Surface. 

16.  Square  of  the  diameter  of  a  sphere  X  3  1416  r=  Surface. 

17.  Square  of  the  circumference  of  a  sphere  X  0-3183  -^  Surface 

18.  Square  root  of  surface  of  a  sphere  X  0'28i095  =  Radius. 

19.  Square  root  of  the  surface  of  a  sphere  X  0*56419  =  Diameter 

20.  Square  root  of  the  surface  of  a  sphere  X  1  •772454  =  Circumference. 

21.  Cube  of  the  diameter  of  a  sphere  X  0-5236  =  Solidity. 

22.  Cube  of  the  radius  of  a  sphere  X  41888  ^  Solidity. 

23.  Cube  of  the  circumference  of  a  sphere  X  0016887  =  Solidity. 

24.  Cube  root  of  solidity  of  a  sphere  X  0-6203505  =  Radius. 

25.  Cube  root  of  the  solidity  of  a  sphere  X  12407  =  Diameter. 

26.  Cube  root  of  the  solidity  of  a  sphere  X  3-8978  =  Circumference. 

27.  Radius  of  a  sphere  X  1-1547  =  Side  of  inscribed  cube. 

28.  Side  of  inscribed  cube  X  0-8660254  =  Radius. 

29.  The  square  of  the  side  of  a  tetraedron  X  1. 7320508  =  Surface. 

30.  The  square  of  the  side  of  a  hexaedron  X  6-0000000  =  Surface. 

31.  The  square  of  the  side  of  an  octaedron  X  3-4641016  =  Surface. 

32.  The  square  of  the  side  of  dodecaedron  X  206457288  =  Surface. 

33.  The  square  of  the  side  of  icosaedron  X  8-6602540  =  Surface. 

34.  The  cube  of  the  side  of  a  tetraedron  X  01 178511  =  Solidity. 

35.  The  cube  of  the  side  of  a  hexaedron  X  1000000  =  Solidity. 

36.  The  cube  of  the  side  of  an  octaedron  X  0-4714045  =  Solidity. 

37.  The  cube  of  the  side  of  a  dodecaedron  X  7-6631189  =  Solidity. 

38.  The  cube  of  the  side  of  an  icosaedron  X  2-181695  =  Solidity. 

A  slight  knowledge  of  the  principles  of  geometry  will 
enable  the  pupil  to  deduce  the  above  multiplies.  For 
illustration,  determine  the  side  of  a  cube  that  is  inscribed 
in  a  sphere. 

c  D  1^6t  X  =  the  side  of  the  in- 

scribed cube.  In  the  right- 
angled  triangle  BFH,  BF 

T 


\ 

M               / 

\ 

F 

\ 

1^ 

^-s/Sk'  +  FH'CseeArt. 
E  240),  or  BF  =  -v/^2- 

In  the  right-angled  trian- 
gle, BDF,  DF  =  X  and  BF 
=  v^2X^  J  therefore,  BD 
=  -v/SX^-  But  BD  =  the 
diameter  of  the  sphere, 
which  we  will  call  unity ; 
^^  hence,  1  =  -s/sx*^'  or  1  = 

X^3  =  X  times  1 73205+;  consequently,  X,  the  side  of  the 


AKT.  264.] 


MENSURATION    OjT    SOLIDS. 


267 


cuLe,  =  y.TT2T!T  of  the  diameter;  or,  •57736-f-timesthe  diarieter 
of  the  sphere ;  and  since  Radius  equals  ^  of  the  diameter ;  X, 
the  side  of  the  cube,  will  equal  Radius  times  (2x  "57736  -|-) 
=  Raclnis  times  1 -154724-. 

In  a  similar  manner  the  pupil  may  deduce  other  multi- 
ples that  he  may  desire,  or  the  teacher  require. 

The  Five  Regular  Bodies. 

Art.  264.  A  regular  body  is  a  solid  bounded  by  a 
certain  number  of  similar  and  equal  plane  figures. 

It  is  proved  in  Solid  Geometry  that  only  three  kinds  of 
equilateral  and  equiangular  plane  figures  joined  together 
can  make  a  solid  angle;  hence  but  Jive  regular  bodies  can 
possibly  be  formed. 


1.  A  re^ra^^r^m  is  a  solid  hav- 
ing four  triangular  faces. 


Rkmark.— If  figures  similar  to  those  annexed  to  the  definitions,  be  drawn 
on  pasteboard,  and  cut  out,  by  cutting  through  the  bounding  lines,  and  if  tlie 
other  lines  be  cut  hall'  through,  and  then  the  parts  be  turned  up  and  glued 
together,  the  bodies  defined  will  be  formed. 


2.  A  Hexaedron,  or  cube,  is  a 
solid  having  six  square  faces. 


. 


3.  An  Odaedron  is  a  regu- 
lar solid  having  eight  trian- 
gular faces. 


268 


MENSURATION. 


[chap.    XI 


4.  A  Doaecae- 
dron  is  a  solid 
having  twelve 
pentagonal  faces. 


5.  An  Icosae- 
dron  is  a  solid 
having  twenty 
triangular  faces. 


Surfaces  of  the  Five  Regular  Bodies. 
Problem  1 — Given  the  side  of  a  tetraedon,  to  find  its  surface. 

Multiply  the  square  of  the  linear  side  by  the  ^/s. 
1.  If  the  side  of  a  tetraedron  is  1,  what  is  its  surface  ? 
'    2.  If  the  side  of  a  tetraedron  is  8  feet,  what  is  its  surface  ? 
Problem  2. — Given  the  side  of  a  hezaedron  to  find  its  surface. 
D/Ftvltijply  the  square  of  the  side  by  6 . 

1.  If  the  side  of  a  hexaedron  is  1,  what  is  its  surface  ? 

2.  If  the  side  of  a  hexaedron  is  4  feet,  what  is  its  surface  ? 

Problem  3. — Given  the  side  of  an  odaedron,  to  find  its  surface 
Multiply  the  square  of  the  side  by  2  ^^/g 

1.  If  the  side  of  an  octaedron  is  1,  what  is  its  surface  ? 

2.  If  the  side  of  an  octaedron  is  8,  what  is  it  surface  ? 

Problem  4. — Given  the  side  of  a  dodecaed^on,  to  find  its  sur- 
face. 


Multiply  15  times  the  square  of  the  side  by  \/l  +  f  ^/5. 


ART.  264.1 


MENSURATION    OF    SOLIDS 


269 


1.  If  the  lineal  side  of  a  dodecaedron  is  1,  what  is  its 
surface  ? 

2,  If  the  side  of  a  dodecaedron  is  9,  what  is  its  surface  ? 

Problem  5. — Given  the  side  of  an  icosaedron.,  to  find  its  surface, 
Multiply  5  times  the  square  of  the  side  by  \/S. 

1.  If  the  side  of  an  icosaedron  is  1,  what  is  its  surface  ? 

2.  What  is  the  surface  of  an  icosaedron,  the  side  of 
which  is  6  feet  ? 

Remark. — From  the  answers  of  the  examples  given  under  the  preceding 
problems,  in  which  the  lineal  side  is  1,  the  following  table  may  be  formed. 

TABLE 

Showing  the  surfaces  of  the  five  regulalr  bodies^  when  the 
linear  side  is  1. 


Number 
of  sides. 

Names  of  bodies. 

Surfaces  of  bodies. 

4 

6 

8 

12 

20 

Tetraedron. 

Hexaedron. 

Octaedron. 

Dodecaedron. 

Icosaedron. 

1-7320508 
6-0000000 
3-4641016 
20-6457288 
8-6602540 

Problem  6.— Given  the  side  of  any  of  the  regular  bodies,  to 
find  its  surface. 

MiiUiply  the  square  of  the  length  of  the  side  by  the  tabular 
area  opposite  the  figure  mentioned. 

1.  The  side  of  a  tetraedron  is  14  in.;  what  is  its  surface  ? 

2.  The  side  of  a  hexaedron  is  7  feet;  what  is  its  surface  ? 

3.  The  side  of  a  tetraedron  is  18  feet;  what  is  its  surface? 

4.  The  side  of  an  octaedron  is  12  inches;  what  is  its 
surface  ? 

5.  The  side  of  a  hexaedron  is  16  feet;  what  is  its  surface  ? 

6.  The  side  of  a  dodecaedron  is  18  inches;  what  is  its 
surface  ? 

7.  The  side  of  an  octaedron  is  10  feet;  what  is  its 
surface  ? 


2^0  MENSURATION.  [cHAP.  XI. 

8.  The  side  of  an  icosaedroa  is  20  inches;  what  is  its 
surface  ? 

9.  The  side  of  a  dodecaedron  is  24  feet;  what  is  its 
surfalce  ? 

10.  The  side  of  an  icosaedron  is  16  feet;  what  is  its 
surface  ? 

Solidity  of  the  Keg-ular  Bodies. 

Problem  1. — Given  the  side  of,  a  tetraedron,  to  find  its 
solidity. 

Multiply  J^  of  the  cube  of  the,  lineal  side  by  the  ,,^2. 

1.  If  the  lineal  side  of  a  tetraedron  is  1,  what  is  its 
solidity  ? 

2.  If  the  side  of  a  tetraedron  is  4  inches,  what  is  its 
solidity  ? 

Problem  2. — Given  the  lineal  side  of  a  hexaedron,  to  find 

its  solidity. 

Cube  the  Side. 

1.  If  the  lineal  side  of  a  hexaedron  is  1,  what  is  its 
solidity  ?  ^ 

2.  If  the  side  of  a  hexaedron  is  6  feet,  what  is  its 
solidity  ? 

Problem  3. — Given  the  side  of  an  octaedron,  to  find  its 
solidity. 

Multiply  the  cube  of  the  side  by  the  ^Ji,  and  \  of  the 
product  will  be  the  solidity. 

1.  What  is  the  solidity  of  an  octaedron,  the  side  of 
which  is  1  ? 

2.  What  is  the  solidity  of  an  octaedron,  the  side  of 
which  is  4  inches  ? 

Problem  4. — Given  the  side  of  a  dodecaedron.  to  find  its 
solidity. 

Add  4:^  to  2l,^y5,  and  divide  this  sum  by  iO;  then  muiti' 
ply  the  square  root  of  this  quotient  by  5  times  the  cube  of  tht 
side. 


ART.  266.J 


PHILOSOPHICAL   PROBLEMS. 


271 


1.  What  is  the  solidity  of  a  dodecaedron,  the  side  of 
which  is  1  ? 

2.  What  is  the  solidity  of  a  dodecaedron,  the  side  of 
which  is  8  inches  ? 

Problem  5 Given  the  side  of  an  icosaedron,  to  find  its 

solidity. 

Divide  the  sum  of  7  aTid  3^5  by  2;  then  multiply  the 
square  root  of  this  quotient  by  f  of  the  cube  of  the  side. 

1.  What  is  the  solidity  of  an  .icosaedron,  the  side  of 
which  is  1  ? 

2.  What  is  the  solidity  of  an  icosaedron,  the  side  of 
which  is  4  inches  ?  '    . 

Remark.— From  the  answers  of  the  examples  given  under  the  preceding 
yrobleins.  in  which  the  lineal  side  is  1,  the  following  table  may  be  formed. 

TABLE. 


Number 
of  Sides. 

4 

6 

8 
12 
20 

Names  of  Bodies. 

Solidity  of  Bodies. 

Tetraedron. 

Hexaedron. 

Octaedron. 

Dodecaedron. 

Icosaedron. 

.1178511 

1.0000000 

.4714045 

7^.6631189 

•    2.1816950 

Problem  6 Given  the  side  of  any  of  the  regular  bodies,  to 

find  its  solidity. 

Multiply  the  cube  of  the  side  by  the  solidity  opposite  the 
given  figure  in  the  above  table. 

1.  What   is  the   solidity  of  the  five   regular   bodies, 
respectively,  the  side  of  each  being  8  inches  ? 

2.  What   is  the   solidity   of  the   five   regular  bodies, 
respectively,  the  side  of  each  being  12  inches. 


PHILOSOPHICAL    PROBLEMS. 

Art.   265.    The  spaces  described  by  bodies  falling  from 
a  state  of  rest  under  the  influence  of  gravity^  are  yropor^ 


2T2  PHILOSOPHICAL   PROBLEMS.  [cHAP.    XI. 

tioned  to  the  squares  of  the  times  during  which  they  art 


Art.  266.  The  spaces  described  by  falling  bodies  are 
also  proportioned  to  the  squares  of  the  velocities  which  they 
acquire  in  falling  over  those  spaces* 

A  body  in  1  second  of  time  will  fall  16^2  ^^^^j  ^^^^  i* 
Id  feet.  The  velocity  acquired  in  the  same  time  is  32 
feet. 

Let  T  equal  the  time,  during  which  a  body  has  been 
falling;  D,  the  distance  it  has  fallen;  and  Y,  the  velocity 
acquired. 

From  the  above  laws  and  facts  we  have  the  following 
proportions,  from  which  the  general  formulas  relating  to 
falling  bodies  may  be  deduced. 

1.  2. 

P :  T2 ::  16  :  D ;  hence,  D=16T2 ;  .-. ,  T=    /^g=]VL 

3.  4. 

32' :  V2 ::  16  :  D ;  hence,D=l';  .-.  ^\=^Mb=^'/b 
64 

Placing  the  right  hand  member    of  1  and  3  equJal  cc 

each  other,  we  have, 

y2 
16  T2  =  —    j — From  which  we  deduce  the  following  : 
64 

^-    '^=32 
6.    V=32T 

Pupils  should  become  familiar  with  the  six  preceding 
formulas,  which  we  will  arrange  differently  for  the  con- 
venience of  reference. 

3.  D=16T2  6.  V==32T 


Olmsted's  Natural  Philosophy. — However,  these  laws  arc  nolstr'M>.iy  t  rreti. 


ART.    268.]  PHILOSOPHICAL   PROBLEMS.  2t3 

1.  How  far  will  a  leaden  ball  fall  in  12  seconds;  14 
seconds;  25  seconds;  and  60  seconds,  resiDectively  ?  (See 
formula  3d.) 

2.  How  long  will  a  body  be  in  falling  1024  feet;  1600 
feet;  10000  feet;  and  722500  feet,  respectively  ?  (See 
formula  1st.) 

3.  In  what  time  would  a  body  acquire  a  velocity  of  128 
feet;  160  feet;  288  feet;  1024  feet;  and  3072  feet,  respec- 
tively ?  (See  formula  2nd.) 

4.  What  velocity  would  a  body  acquire  in  4;  7;  9;  12; 
25;  and  60  seconds,  respectively?  (See  formula  6th.) 

5.  What  velocity  would  a  body  acquire  in  falling  1024 
feet;  7225  feet;  625  feet;  3025  feet;  and  9025  feet, 
respectively  ?  (See  formula  5th.) 

6.  Through  what  space  would  a  body  have  fallen  to  ac- 
quire a  velocity  of  96  feet;  192  feet;  768  feet;  288  feet; 
and  384  feet,  respectively  ?  (See  formula  4th.) 

Art.  267^.  The  time  of  tlie  vibrations  of  pendulums  are 
to  each  other  as  the  square  roots  of  their  lengths  ;  hence,  their 
lengths  are  as  the  squares  of  their  times  of  vibration. 

A  pendulum  that  vibrates  seconds  is  39^  inches  in 
length. 

1.  What  is  the  length  of  a  pendulum  that  shall  vibrate 
3  times  a  second  ? 

2.  What  is  the  length  of  a^  pendulum  that  shall  vibrate 
once  in  5  seconds  ? 

3.  What  is  the  length  of  a  pendulum  that  shall  vibrate 
once  in  a  minute  ? 

4.  How  often  will  a  pendulum  vibrate,  the  length  of 
which  is  225  inches  ? 

5.  How  often  will  a  pendulum  vibrate,  the  length  of 
which  is  144  feet  ? 

Art.  268.  The  gravity  of  any  body  above  the  earth's 
surface  decreases,  as  the  squares  of  its  distance,  in  semi- 
diameters  of  the  earth,  from  its  centre  increases.     Hence, 

Th£.  weight  of  a  body  on  the  earthUs  surface,  is  to  its 
weight  at  any  <issignable  distance  above  the  surface  of  thi 

12* 


274  MISCELLANEOUS    QUESTIONS.  [CAAP.    XI 

earth;  as  the  square  of  its  distance  from   the  ear  til's  centre, 
to  the  square  of  the  earthUs  semidiameter ,  and  vice  versa. 

1.  If  a  body  weigh  75'0  pounds  at  the  earth's  surface,^ 
how  much  would  it  weigh  20000  miles  above  its  surface  ? 

2.  If  a  body  at  the  earth's  surface  weigh  3600  pounds, 
how  much  would  it  weigh  240000  miles  above  its  centre, 
the  distance  of  the  moon  from  the  earth  ? 

If  a  body  at  the  earth's  surface  weighed  1800  pounds 
but  being  carried  to  a  certain  height  weighs  only  200 
pounds,  what  is  that  height  ? 


MISCELLANEOUS    QUESTIONS. 

Rkmark. — Analvsis  has  been  so  extensively  treated  of  in  the  -'American 
Intellectual"  and  the  "  Practical  Arithmetic,"  that  it  is  deemed  unnecessary 
to  add  anything  more  to  what  has  already  been  givenin  the  preceding  pages. 

1.  A  gentleman,  after  losing  ^  of  all  his  money,  had 
$368  remaining.     How  much  had  he  at  first  ? 

2.  Thomas  has  364  sheep  more  than  James,  and  they 
together  have  1588.     How  many  have  they  respectively  ? 

3.  Henry,  Perry,  and  John,  found  a  purse  containing 
$768,  which  they  agree  to  share  in  proportion  to  the 
numbers  3,  4,  and  5.     How  much  should  each  receive  ? 

4.  A  farmer  gave  to  a  certain  number  of  laborers,  $14 
apiece,  if  he  had  given  them  $19  apiece  it  would  have 
taken  $125  more.     How  many  laborers  were  there  ? 

5.  A  fish-pole,  the  length  of  which  was  24  feet,  was 
broken  into  two  pieces;  and  f  of  the  lengtii  of  the  longer 
piece  equalled  the  length  of  the  shorter.  What  was  the 
length  of  each  piece  ? 

6.  There  is  a  fish  whose  head  is  18  inches  long,  and 
whose  tail  is  as  long  as  its  head  +  f  of  the  length  of  its 
body,  and  whose  body  is  as  long  as  its  head  and  tail  both. 
What  is  the  length  of  the  fish  V 

7.  Henry  is  18  years  old,  and  Harvey  is  14;  how  many 
years  since  was  Henry  twice  as  old  as  Harvey  ? 

8.  A  and  B,  together,  have  $8645,  but  A  has  $155 
more  than  3  times  as  much  as  B.  How  many  dollars  has 
each  ? 


ART.    208. J  MISCELLANEOUS    QUESTIONS.  2"75 

9.  A  gentleman  bought  a  hat,  a  vest,  and  a  pair  of 
pants.  The  hat  cost  $6.  The  hat  and  vest  cost  iwice  aa 
much  as  the  pauts.  and  the  hat  and  pants  cost  3  times  as 
much  as  the  vest.  What  was  the  cost  of  the  vest  and 
pants,  respectively  ? 

10.  A  can  earn  a  certain  sum  of  money  in  20  days ;  A 
and  B  together,  can  earn  tlie  same  sum  in  6  days.  How 
long  will  it  take  B  alone  to  earn  the  same  sum  ? 

11.  A  merchant  bought  a  c<n-tain  number  of  yards  of 
cloth  for  $245,  and  after  usin^-  ;)  yards  of  it  himself,  sold 
f  of  the  remainder  for  $99,  which  was  $24  more  than  it 
oost.     How  many  yards  did  he  bny  at  first  ? 

12.  A  person  at  a  game  of  cards  lost  f  of  all  his  money, 
and  then  won  $144;  he  now  lost  ^  of  all  the  money  he 
had,  and  found  he  had  but  $95  remaining.  How  much 
bad  he  at  first  ? 

13.  There  is  a  cask  containing  brandy  and  water;  f  of 
the  whole,  +  12  gallons  is  water;  and  ^  of  the  whole  + 
8  gallons  is  brandy.     How  many  gallons  of  each  ? 

14.  Two  brothers,  James  and  Henry,  have  the  same 
income.  James  contracts  an  annual  debt,  amounting  to 
$165;  Henry  lives  on  -|  of  his  income,  and  saves  yearly 
$101,  after  lending  James  enough  to  pay  his  debt.  How 
much  was  the  yearly  income  of  each  ? 

15.  Two  masons,  A  and  B,  together  can  do  a  certain 
piece  of  work  in  10  days  ;  how  long  would  it  take  each 
separately  to  do  it  providing  A  does  3  times  as  much 
as  B  ? 

16.  If  A  and  B  can,  together,  do  a  certain  piece  of  work 
in  5  days  ;  A  and  C,  in  6  days  ;  and  B  and  C,  in  H  ; 
how  many  days  would  it  require  for  each  to  perform  the 
work  alone  ? 

17.  Divide  $4760  among  three  persons,  James,  William 
and  Mary,  so  that  James'  part  shall  be  to  William's  as  2 
to  3,  and  that  Mary  shall  have  as  much  as  James  and 
William  together  lacking  $920. 

18.  A  and  B  can,  together,  do  a  certain  piece  of  work 
in  10  dys.  ;  A  and  C,  in  15  dys.;  and  B  and  C,  in  20  dys 
In  hi">w  many  days  could  each  perform  the  work  alone  ? 


2T6  MISCELLANEOUS    QUESTIONS.  [cHAP.    XL 

19.  A  gentleman  distributed  a  certain  number  of  dol- 
lars among  four  poor  women  in  the  following  manner  : — ■ 
to  the  first  he  gave  ^  of  the  number  of  dollars  he  had  + 
$1;  to  the  second  ^  the  remainder  -f  $i  ;  in  the  same 
manner  he  gave  to  the  third  and  the  fourth  ;  and  found 
he  had~  yet  one  dollar  remaining.  How  many  dollars 
had  he  at  first,  and  how  much  did  he  give  to  each 
woman  ? 

20.  An  estate  of  $12850  was  was  left  to  four  brothers, 
who  are  17,  15,  13,  and  9  years  of  age,  to  be  so  divided 
that  the  respective  parts,  being  placed  out  at  5  per  cent, 
simple  interest,  should  amount  to  equal  sums  when  they 
become  21  years  of  age,  respectively.  How  much  was 
each  one's  share  ? 

21.  A  man  bought  a  horse  for  $102,  which  was  ^  of 
twice  as  much  as  he  sold  it  for,  lacking  $2.  How  much 
did  he  gain  by  the  bargain  ? 

22.  A  woman  bought  60  oranges.  For  f  of  them  she 
paid  5  cents  for  3  oranges  ;  and  for  the  remainder  3  cents 
for  five  ;  for  how  much  must  she  sell  them  apiece  to  gain 
331  per  cent.  ? 

23.  A  farmer  paid  to  four  of  his  hired  men  tt}  bushels 
of  wheat.  The  first  earned  1  bushel  as  often  as  the  other 
three  earned  |,  f ,  and  |  of  a  bushel,  respectively.  How 
many  bushels  should  each  receive  ? 

^4.  Two  men  A  and  B  were  playing  cards;  B  lost  $84, 
which  was  -^-^  times  |  as  much  as  A  then  had.  When  they 
commenced  playing,  |  of  A's  money  equalled  f  of  B's  ; 
how  much  had  each  when  they  began  to  play  ? 

25.  What  is  the  interest  on  $685-95  from  April  14th  to 
Sept.  19th? 

26.  What  is  the  amount  of  $684-99  from  April  9th, 
1853,  to  July  8th,  1854? 

2*1.  A  has  with  B  the  following  account : — 

1853.  ,  Dr.      I  1853.  Cr. 

March  12th,  Due      .     P45-45  |  Sept.  16th,  Due.   .     $784-50 

At  what  time  is  the  balance  of  the  account  due  ? 

28.  I  sold  t/ie  following  bills  of  goods,  on  tlie  conditions 
below  stated : 


ART.    268.]  MISCELLANEOUS    QUESTIONS  277 

March    6,  1853,  a  bill  amounting  to  $480  on  4  months'  credit. 
Arxril    15,     "  "  "  $-670  on  6       "  "     ■ 

May      25,     "  "  "  iffTBo  on  4       "    ' 

June     28,     "  "  "  $670  on  3       "  " 

How  much  money  will  balance  the  account  July  20th  ? 

29.  Three  farmers,  A,  B,  and  C,  together  have  1920 
acres  of  land;  A  has  40  acres  more  than  B;  and  C  has  as 
many  as  A  and  B  together,  lacking  32.  How  many  acres 
has  each  ? 

30.  What  is  the  discount  on  $847-50  from  May  12th, 
1852,  to  July  25th,  1854  ? 

31.  What  sum  of  money  will  give  $18490  interest  from 
June,  16th,  1853,  to  Sept.  18th,  1854? 

32.  If  A  can  do  a  certain  piece  of  work  in  80  days,  and 
with  the  assistance  of  C,  in  34f  days;  how  long  will  it 
take  C  to  do  the  work  alone  ? 

33.  Three  carpenters.  A,  B,  and  C,  earn  a  certain  sura 
of  money  in  24  days;  A  and  B  can  earn  the  same  amount 
in  48  days;  and  A  and  C,  in  36  days.  How  long  would 
it  take  each  separately  to  earn  the  same  amount  ? 

34.  An  individual  being  requested  to  buy  a*  certain 
number  of  pounds  of  meat,  found,  if  he  bought  beef,  at 
11^  cts.  a  pound,  he  would  have  90  cts.  remaining;  but  if 
he  bought  pork,  at  17|  cts.  a  pound,  he  would  lack  15  cts. 
of  having  money  enough  to  pay  for  it.  How  many  pounds 
of  meat  was  he  requested  to  buy,  and  how  much  money 
did  he  have  ? 

35.  A  lady  bought  a  certain  number  of  apples,  at  the 
rate  of  5  for  2  cents;  and  paid  for  them  with  oranges,  at 
the  rate  of  3  for  2  cents.  How  many  apples  did  she  buy, 
providing  it  took  144  oranges  to  pay  for  them  ? 

36.  Three  farmers,  Thomas,  William,  and  Henry,  talking 
of  their  sheep;  says  Thomas  to  William,  I  have  4  times  as 
many  sheep  as  you;  says  William  to  Henry,  I  have  |  as 
many  as  you;  and  says  Henry  to  Thomas,  if  I  had  63  sheep 
more,  I  should  then  have  as  many  as  you.  How  many 
iiad  each  ? 

37.  A  man  was  hired  for  160  days,  on  this  condition: 
that  for,  every  day  he  worked,  he  should  receive  $144, 
and  for  every  day  he  was  idle,  he  should  pay  96  cents  foi 


2T8  MISCELLANEOUS    QUESTIONS.  [cHAP.  XL 

his  board.    At  the  expiration  of  the  time  he  received  $64 
How  many  days  did  he  work  ? 

38  A,  B,  and  C  formed  a  co-partnership  :  A  advanced 
$10000;  B  $8000;  and  C  $7000.  At  the  end  of  6 
months,  A  withdrew  $3000  from  the  business;  B  with- 
drew $1500;  and  C  increased  his  stock  by  ^  of  its 
original  amount.  At  the  end  of  the  year,  they  had  gained 
$5584'60.     How  much  should  each  receive  ? 

39.  A  gentleman  willed  $8640  to  his  wife,  son,  and 
daughter,  to  be  divided  among  them  in  the  proportion  of 
I,  I  and  |.  The  widow  dying  soon  after,  the  whole  sum 
was  divided  in  due  proportion  between  the  two  children. 
How  much  did  each  receive  ? 

40.  A  cistern  receives  water  from  3  pipes;  the  first  of 
which  would  fill  it  in  12  hours;  the  second  in  8  hours; 
and  the  third  in  6  hours.  In  what  time  v/ould  these 
three  pipes  together  fill  the  cistern,  providing  i  of  the 
whole  capacity  of  the  cistern  leaked  out  in  each  hour  ? 

41.  A  merchant  spent  |  of  his  money  for  silks;  |  of 
the  remainder  for  dry  goods;  |  of  the  remainder  for 
groceries;  and  the  remainder,  which  was  $28t"65,  for 
stationery.     How  much  money  did  he  expend  in  all  ? 

42.  Four  men  contracted  to  grade  a  turnpike  road  for 
$12000.  In  accomplishing  the  work,  one  of  the  men 
furnished  45  laborers  for  74  days;  another,  54  laborers 
for  66  days;  another,  75  laborers  for  84.  days;  and  the 
other,  95  laborers  for  85  days.  How  much  should  each 
contractor  receive  ? 

43.  An  agent  receives  $5685  to  invest  in  merchandise, 
at  a  commission  of  1|  per  cent,  on  the  amount  of  purchase 
that  can  be  made  after  his  percentuni  is  deducted.  What 
is  the  amount  of  purchase;  also,  his  commission  ? 

44.  An  upholsterer  realized  a  profit  of  25  per  cent, 
by  selling  carpeting,  at  $1.50  a  yard.  What  would 
have  been  the  loss  per  cent,  if  he  had  sold  it  at  $0*80  a 
yard  ? 

45.  How  much  would  a  person  gain  or  lose  by  borrow- 
ing $2000  from  May  12th,  1852,  to  Nov.  12th,  1854,  at 
7  per  cent,  and  lending  the  same  sum,  at  6^^  per  cent., 


ART.  26'8.]  MISCELLANEOUS    QUESTIONS.  219 

and  on  such  conditions  as  will  enable  him  to  compound 
the  interest  every  6  months  ? 

46.  A  drover  bought  288  head  of  cattle,  at  $42f  a 
head,  and  pays  for  them  with  the  proceeds  of  a  note 
which  is  discounted  in  a  bank  for  90  days,  at  7  per  cent. 
At  the  end  of  25  days,  he  sells  the  cattle,  at  $68|-  a  head, 
and  puts  the  proceeds  on  interest,  at  8f  per  cent.,  until 
his  note  is  to  be  paid  at  the  bank.  What  profit  does  he 
make  by  these  transactions,  after  paying  $374'65  for 
the  cattle  while  he  had  them  ? 

47.  A  merchant  took  a  farmer's  note  for  $585-50,  due, 
without  interest.  May  14th,  1853.  Some  time  afterwards, 
the  farmer  got  possession  of  a  note  against  the  merchant 
for  $894-85,  due,  without  interest,  Nov.  25th.  When,  in 
equity,  ought  the  balance  to  be  paid  ?  Suppose  money  to 
be  worth  7' per  cent.,  and  they  desire  to  settle  Aug.  15th j 
how  stands  the  matter  of  debt  between  them  ? 

48.  A  is  indebted  to  B  $885;  $125  of  which  is  due 
May  4th;  $244,  June  18th;  $345,  Aug.  12th;  and  the 
remainder,  Oct.  25th, — without  interest.  At  what  time 
might  the  whole,  in  equity,  be  paid  at  once  ? 

49.  What  must  be  the  dimensions  of  a  granary  which 
shall  contain  2400  bushels  of  wheat;  its  length  to  be 
twice  its  breadth,  and  its  breadth  and  height  equal  ? 

50.  What  is  the  difference  in  arm  between  two  fields 
of  the  same  perimeter;  one  of  which  is  a  square,  and  the 
other  85  rds.  long,  and  251  rods  wide  ? 

51.  An  individual  was  requested  to  purchase  1084 
bushels  of  grain,  consisting  of  rye,  wheat,  and  barley; 
I  of  the  number  of  bushels  of  rye  was  to  equal  \  of  the 
number  of  bushels  of  wheat,  and  |  of  the  number  of 
bushels  of  wheat  was  to  equal  f  of  the  number  of  bushels 
of  barley.  How  many  bushels  of  each  kind  must  he 
buy? 

52.  A  man  being  asked  the  hour  of  the  day,  replied, 
that  I  of  the  time  past  noon  equalled  ^  of  the  time  from 
now  to  midnight.     What  was  the  time  't 

53.  A  tree,  whose  length  was  156  feet,  was  broken  into 
two  pieces  by  falling;  1^  times  the  length  of  the  top  piece, 


280  MISCELLANEOUS    QUESTIONS.  [cHAP.    XI. 

equals  l}  the  bottom  piece,  +  12  feet.    What  is  the  length 
of  the  two  pieces  respectively  ? 

54.  A  man  bought  a  cow,  a  horse,  and  an  ox  for  $350. 
For  the  horse  he  gave  4  times  as  much  as  for  the  ox,  lack- 
ing $40;  and  for  the  ox,  twice  as  much  as  for  the  cow, 
lacking  $12,     What  did  he  give  for  each  ? 

55.  A  farmer  has  299  sheep  in  two  different  fields;  the 
number  in  the  first  field  equals  If  times  the  number  in  the 
second  field,  +  48.     How  many  are  there  in  each  field  ? 

56.  An  individual,  after  spending  |  of  all  his  money, 
and  f  of  what  remained,  lacking  $12^,  had  only  $347^ 
remaining.     How  much  had  he  at  first  ? 

5t.  There  is  an  island  36  miles  in  circumference,  and 
three  men.  A,  B,  and  C  start  from  the  same  point,  and 
travel  the  same  way  around  it;  A  4  miles  an  hour;  B,  12; 
and  C,  20.  In  what  time  will  they  all  be  together;  and 
in  what  time  will  they  all  meet  at  the  place  from  which 
they  started  ? 

58.  A  note  of  $1200,  given  Feb.  3d.,  1851,  has  received 
the  following  indorsements  :  March  12th,  1852,  indorsed 
$365-45;  Nov.  14th,  1852,  indorsed  $285-90;  Jan.  12th, 
1853,  indorsed  $484-12|.  How  much  remains  due  March 
20th,  1854,  interest  computed  at  1  per  cent.  ? 

59.  Four  masons.  A,  B,  C,  and  D  engage  to  build  a 
certain  piece  of  wall  for  $660.  While  A  can  build  5  rods, 
B  can  build  1^,  C  3|,  and  D  6i.  When  the  wall  is  |  com- 
pleted, D  ceases  to  labor  upon  it,  and  A,  B,  and  C  finish 
it.     How  much  should  each  receive  ? 

60.  A  market-woman  bought  oranges,  at  10  cents  a 
dozen,  half  of  which  she  exchanged  for  lemons,  at  the  rate 
of  9  oranges  for  T  lemons;  she  then  sold  all  her  oranges 
and  lemons,  at  1|  cents  apiece,  and  thereby  gained  24 
cents.  How  many  oranges  did  she  buy,  and  how  much 
did  they  cost  ? 

61.  If  A  can  perform  a  certain  piece  of  work  in  |  of  a 
"Hay;  B  |  of  a  day;  and  C  in  y^  of  a  day;  how  many  times 

longer  will  it  take  C  to  do  the  work  alone,  than  it  will 
take  A  and  B  together  to  do  it  ? 

62.  A  traveler  had  stolen  from  him  {--J  of  all  his  money: 


ART.    268.]  MISCELLANEOUS   QUESTIONS.  281 

the  thief  was  caught,  but  not  until  he  had  spent  f  of  it, 
the  remainder,  $647-37|^,  was  given  back.  How  much 
money  had  the  traveler  at  first  ? 

63.  Three  men,  A,  B,  and  C,  built,  a  stone  wall  :  A 
built  15  rods;  B  built  as  much  as  A;  f  as  much  as  C;  and 
C  built  as  much  as  A  and  B  together,  lacking  5  rods. 
How  many  rods  did  they  all  build,  and  how  many  did 
B  and  C  build,  respectively  ? 

64.  At  what  time  between  2  and  3  o'clock  will  the  hour 
and  minute  hands  of  a  clock  be  together  ? 

65.  A  person  being  asked  his  age,  replied,  that  if  his 
age  were  increased  by  its  f ,  its  f ,  and  25^  years  more,  the 
sum  would  equal  3^  times  his  age.     What  was  his  age  ? 

66.  A  person  being  asked  the  time  of  day,  replied,  that 
I  of  the  time  past  noon,  equal  f  of  the  time  from  then  to 
midnight,  lacking  12  minutes  and  36  seconds.  What  was 
the  time  ? 

61.  When  James  was  married,  he  was  3  times  as  old  as 
his  wife,  but  when  they  had  been  married  60  years,  he 
was  only  1|  times  as  old.  How  old  was  each  when  they 
were  married  ? 

68.  Four  individuals  found  a  purse,  containing  $2445, 
which  they  agree  to  share  in  the  proportion  of  f,  |,  f, 
and  ^.     How  much  should  each  receive  ? 

69.  A  deer  is  a  180  leaps  before  a  hound,  and  takes 
4  leaps  to  the  hound's  9 ;  and  5  of  the  deer's  leaps  are  equal 
to  9  of  the  hound's.  Hov/  many  leaps  must  the  hound 
take  to  catch  the  deer  ? 

10.  A  market-woman  bought  a  certain  number  of  pine- 
apples, at  the  rate  of  3  for  40  cents,  and  as  many  more  at 
the  rate  of  5  for  45  cents;  and  sold  them  all  at  the  rate 
of  7  for  81i  cents,  and  thereby  gained  $1-60,  Hew  many 
pine-apples  did  she  buy  ? 

71.  A  boy  bought  a  certain  number  of  apples  at  the 
rate  of  4  for  a  cent,  and  as  many  more  at  5  for  a  cent  ; 
and  sold  them  out,  at  the  rate  of  9  for  5  cents,  and  by  so 
doing  gained  45  cents.     How  many  apples  did  he  buy  ? 

72.  A  mechanic  and  his  two  sons  earned  $1490  in  1 
year  ;  the  father  earned  twice  as  much  as  the  elder  son^ 


282  MISCELLANEOUS    QUESTIONS.  [cHAP.    XI. 

lacking  $tO,  and  the  younger  son  earned  i  as  much  as  the 
elder  son -f- 160  dollars.     How  much  did  each  earn  ? 

Y3.  A  woman  bought  a  certain  number  of  oranges,  at 
the  rate  of  5  for  3  cents,  as  many  more  at  the  rate  of  7 
for  5  cents  ;  and  sold  them  all,  at  the  rate  of  15  for  11 
cents,  and  thereby  gained  25  cents.  How  many  oranges 
did  she  buy  ? 

74.  A  merchant  bought  three  pieces  of  cloth  for  $639: 
f  of  the  cost  of  the  first  piece  equals  -|  of  the  cost  of  the 
second;  and  f  of  the  cost  of  the  second  piece  equals  |  of 
the  cost  of  the  third.     How  much  did  each  piece  cost  ? 

75.  A  merchant  bought  three  pieces  of  cloth  ;  the  first 
piece  contained  |  as  much  as  the  second  piece  +  12  yards; 
and  I  of  the  number  of  yards  in  tlie  second  piece  equaled 
f  of  the  number  of  yards  in  the  third.  How  many  yards 
in  each  piece,  providing  there  were  8  yards  more  in  the 
third  piece  than  in  the  second  ? 

76.  It  is  found  that  f  of  A's  +  -^  of  B's  fortune 
equals  $5400  ;  and  that"*!  of  A's  fortune  equals  1|  times 
I  of  B's +  $24.     What  is  the  fortune  of  each  ? 

77.  A  hound  ran  150  rods  before  he  caught  a  hare;  and 
■j?3  the  distance  the  hare  ran  before  it  was  caught  equal- 
ed the  distance  it  was  a-head  v»'hen  they  started.  How 
far  after  the  chase  commenced,  did  the  hare  run  before  it 
was  caught  ? 

78.  A  and  B  started  from  the  same  point,  and  ran  in 
the  same  direction  ;  B  ran  132  rods;  then  /g  the  distance 
A  had  run  equaled  the  distance  A  was  in  advance  of  B. 
How  much  did  A  gain  on  B  in  running  132  rods  ? 

79.  A  gentleman  left  his  son  a  fortune  ;  ^  of  which  he 
spent  in  2  years  ;  ^  of  the  remainder  lasted  him  3  years 
longer  ;  f  of  the  remainder  lasted  him  5  years  longer  when 
he  had  only  $784912^  left.  How  much  did  his  father 
leave  him  ? 

80.  I  of  A's  number  of  sheep  is  to  f  of  B  as  |  to  f  ; 
and  f  of  B's  number  +  |  of  A's  equals  360.  How  many 
sheep  has  each  ? 

81.  Find  the  fortunes  of  A,  B,  C,  D,  E  and  F,  by  know- 
ing that  B  is  worth  $220,  which  is  ^  as  much  as  A  and  0 


ART.  268.]  MISCELLANEOUS    QUESTIONS.  283 

are  worth,  and  that  A  is  worth  ^  as  much  as  B  and  C  ; 
and  also,  that,  if  76  times  the  sum  of  A's,  B's,  and  C's  for- 
tune were  divided  in  the  proportion  of  f ,  ^,  and  },  it 
would,  respectively,  give  |  of  D's,  |  of  E's,  and  f  ot  F's 
fortune. 

82.  There  is  a  park  16  rods  square,  and  it  is  desired  to 
make  a  gravel  walk  around  it  that  shall  contain  J-f  of  the 
whole  area  of  the  park.  What  should  be  the  width  of  the 
gravel  walk  ? 

83.  A  speculator  sold  flour  at  $5  a  barrel  ;  }  of  which 
equaled  his  gain.  How-much  would  he  have  gained  per 
cent,  if  he  had  sold  it  at  $6-25  a  barrel  ? 

84.  A  merchant  sold  a  quantity  of  goods  for  $6*184  ; 
and  thereby  cleared  y^  of  this  money.  If  he  had  sold 
them  for  $6999,  what  would  he  have  gained  per  cent.  ? 

85.  A  speculator  sold  a  quantity  of  cotton  for  $8484  ; 
and  by  so  doing  gained  }  of  what  it  cost  him.  How  much 
would  he  have  gained  per  cent,  if  he  had  sold  it  for  $9898  ? 

86.  A  gentleman  bought  f  of  a  farm  for  $9000  ;  and 
sold  to  B  i  of  his  share  ;  B  sold  to  C  |  of  what  he  re- 
ceived ;  C  sold  to  D  I  of  what  he  received  ;  and  D  sold 
to  E  f  of  what  he  received.  What  part  of  the  farm  did 
each  man  buy,  and  how  much  did  it  cost  him  ? 

87.  I  bought  f  of  a  house,  valued  at  $18000  ;  and  sold 
^  of  my  share  to  A  ;  A  sold  |  of  his  share  to  B  ;  and  B 
sold  I  of  his  share  to  C.  What  part  of  the  value  of  the 
house  does  each  own,  and  how  much  does  C  pay  for  his 
part  ? 

88.  An  individual  sold  two  horses,  at  $630  apiece  ;  for 
one  he  received  25  per  cent,  more  than  its  value,  and  for 
the  other  25  per  cent,  less  than  its  value.  Did  he  gain  or 
lose  by  the  bargain,  and  how  much  ? 

89.  B's  fortune  added  to  |  of  A's,  which  is  to  B's  as  2 
to  3,  being  put  on  interest  for  6  years,  at  8  per  cent,, 
amounts  to  $988.     What  is  the  fortune  of  each  ? 

90.  How  much  grain  must  a  farmer  take  to  mill,  that 
he  may  *'etch' away  14*4  bushels,  after  Ihe  miller  has  taken 
ty\  per  cent,  of  all  he  took  there  ? 

91.  The  interest  on  the  sum  of  i  of  A's  +  |  of  B's 


284  MISCELLANEOUS    QUESTIONS.  [cHAP.    XL 

money  for  4  years,  at  6  per  cent.,  is  $480.  What  is  tlie 
fortunf^  of  each,  providing  i  of  B's  money  equals  3  times 
I  of  A's  ? 

92.  The  amount  of  |  of  A's  fortune  +  f  of  B's  for  two 
years,  at  5  per  cent.,  is  $4950.  What  is  the  fortune  of 
each,  providing  ^  of  A's  money  equals  only  f  of  f  of  B's  ? 

93.  If  the  interest  on  the  sum  of  A's  and  B's  fortune 
for  1  years  and  6  months,  at  4  per  cent.,  is  $3213  ;  and 
I  of  A's  fortune  equals  |  of  B's ;  what  is  the  fortune  of 
each  ? 

94.  What  will  be  the  result,  if  from  the  sum  of  3,  f 

31 
31,  I  of  3,  3i  of  3-},  4  of  ^  we  subtract  the  sum  of  i,  | 

2—  2—        3— 

of  i,  1  of  31  ;  ^  of  5,  ~  of -gT,    and  3  j;  multiply  this 

difference  by  the  greatest  common  divisor  of  315  and 
405;  divide  this  product  by  the  least  common  multiple  of 
6,  9,  and  24;  reduce  the  quotient  to  its  lowest  terms;  add 
1  of  I  to  the  result;  multiply  |   of  this  sum  by  2^;   and 

divide  the  product  by  i  of  i  of  4i  of  f  |  of  ^f  ? 

95.  Divide  $3106-50  among  A,  B,  C,  and  D,  in  the 
following  proportion  : — A,  B,  and  C  are  to  have  |^  of  it; 
B,  C,  and  D  are  to  have  |^  of  it;  A,  C,  and  D  are  to 
have  y'^o  of  it;  and  A,  B,  and  D  are  to  have  f  of  it. 
According  to  the  above  estimates,  how  much  ought  each 
to  receive  ? 

96.  An  individual,  for  two  successive  years,  spent  f 
more  than  his  yearly  income;  and  found  that,  in  6  years, 
by  saving  -^^  of  his .  annual  income,  he  was  able  to 
discharge  the  debt,  and  have  $80  remaining.  What  was 
his  annual  income  ? 

91.  How  many  cannon  balls,  8  inches  in  diameter, 
can  be  put  into  a  cubical  vessel,  2  feet  on  a  side;  and 
how  many  gallons  of  wine  will  it  contain  after  it  is  filled 
with  balls,  allowing  the  balls  to  be  hollow,  the  hollow 
being  4  inches  in  diameter,  and  the  opening  leading  to  it, 
to  contain  1^  solid  inches  ? 


ART.  268.]  MISCELLANEOUS  QUESTIONS.  285 

98.  A  farmer  sold  hay,  at  $10-50  a  ton,  aud  cleared 
1  of  his  money;  but  hay  growing  scarce,  he  sold  it, 
at  $12  a  ton.  What  did  he  clear  per  cent,  by  the  latter 
price  ? 

99.  If  24  men,  in  132  days  of  9  hours  each,  dig  a 
trench  that  is  4  degrees  of  hardness,  337|  feet  long,  5f 
feet  wide,  and  3^  feet  deep;  how  many  men  will  be 
required  to  dig  a  trench  that  is  t  degrees  of  hardness, 
2321  feet  long,  3|  feet  wide,  and  2i  feet  deep,  in  5i  days 
of  11  hours  each  ? 

100.  From  a  certain  sum  of  money  I  took  away  its  |, 
ajid  in  its  stead  placed  $200;  I  then  took  from  this  sum 
its  i,  and  in  its  stead  placed  $100;  I  now  took  away  its 
f,  and  found  I  had  only  $480  left.  How  much  was  the 
original  sum  ? 

101.  From  a  sum  money,  $360  more  than  its  }  was 
taken  away;  from  the  remainder,  $280  more- than  its  i 
was  taken  away;  and,  from  what  now  remained,  $80 
more  than  its  |  was  taken  away,  and  then  there  remained 
only  $80.     What  was  the  original  sum  ? 

102.  From  a  certain  sum  of  money  I  took  its  -i,  and 
put  in  its  stead  $460;  from  the  remainder  I  took  its  |, 
and  put  in  its  stead  $600;  and  from  what  then  remained 
I  took  its  i,  and  put  in  its  stead  $840,  and  found  1  had 
twice  as  much  money  as  I  had  at  first.  How  much  had 
I  at  first  ? 

103.  Make  the  sura,  difference,  product,  and  quotient 
of  15  and  45  the  numerators  of  fractions  which  shalLhave 
^i5,  40,  750,  and  60  for  denominators;  reduce  them  to 
equivalent  fractions  having  a  common  denominator  ;  sub- 
tract the  sum  of  the  last  two  fractions  from  the  sum  of 
the  first  two;  multiply  this  difference  by  the  first  fraction; 
divide  the  product  by  the  greatest  common  divisor  of  the 
numerators;  multiply  the  quotient  by  the  least  common 
muttiple  of  the  denominators;  add  the  first  fraction 
reduced  to  a  decimal  to  this  quotient;  subtract  the  second 
fraction  reduced  to  a  decimal  from  this  sum;  multiply 
this  remainder  by  the  third  fraction  reduced  to  a  decimal; 
divide  this  product  by  the  fourth  reduced  to  a  decimal; 


286  MISCELLANEOUS  QUESTIONS.  [cHAP.  XI. 

then  reduce  the  quotient  to  a  vulgar  fraction.     What  is 
the  result  ? 

104.  A  merchant  sold  3  pieces  of  broadcloth,  each 
piece  containing  27  yards,  at  $7  a  yard,  on  2  months' 
credit,  and  made  12  per  cent,  on  the  first  cost, — it  had 
been  on  hand  3  months;  7  pipes  of  wine,  at  $4'50  per 
gallon,  at  an  advance  of  18  per  cent,  on  the  first  cost, 
which  had  been  7  months  on  hand, — for  which  he  gave  a 
credit  of  3  months;  and  7  bales  of  cotton,  at  11^  cents  a 
pound,  each  bale  containing  230  pounds,  which  had  been 
on  hand  1  month  and  15  days,  aj;  an  advance  of  20  per 
cent,  on  the  first  cost, — for  which  he  gave  6  months 
credit.  How  much  did  he  make  by  the  operation,  and 
how  much  did  he  make  on  each  article  ? 

105.  Suppose  premiums,  of  three  grades,  to  the  amount 
of  $24  are  to  be  distributed  among  the  pupils  of  a  school. 
The  value  of  a  premium  of  the  first  grade  is  twice  the  value 
of  one  of  the  second  grade;  the  value  of  one  of  the  second 
grade  is  twice  the  value  of  one  of  the  third  grade;  and 
there  are  6  of  the  first  grade,  12  of  the  second,  and  6  of 
the  third.    What  is  the  value  of  a  premium  of  each  grade  ? 

106.  Four  carpenters  built  a  house  in  company.  The 
lot  on  which  they  built  it  cost  $1000;  the  lumber  and 
building  materials  of  all  kinds  cost  $6500;  they  paid  for 
mason-work  $500;  and  for  painting  and  glazing  $350, 
Of  these  expenses  A  paid  |,  B  |,  C  },  and  D  the  residue. 
A  worked  on  the  house  45  days,  at  $1'50  a  day,  with  3 
apprentices,  each  $0*75  a  day  ;  B  worked  75  days,  at 
$1*75  a  day,  with  2  journeyman,  each  $1*25  a  day;  C 
worked  60  days,  at  $r62i  a  day,  with  1  journeyman,  at 
$1'37^  a  day,  and  2  apprentices,  each  $087^  a  day;  and 
D,  the  master  workman,  worked  90  days,  at  $2'25  a  day, 
with  2  journeyman,  each  $1*75  a  day,  and  2  apprentices, 
each  $125  a  day.  The  house  being  completed  it  was  sold 
for  $2500  more  than  it  cost.  How  much  in  equity  ought 
each  partner  to  receive  ? 

107.  A  deer  starts  40  rods  before  a  hound,  and  is  not 
perceived  by  him  until  40  seconds  afterwards;  the  deer 
runs,  at  the  rate  of  10  miles  an  hour;  and  the  hound  after 


ART.    268.]  MISCELLANEOUS    QUESTIONS.  28t 

it,  at  the  rate  of  18  miles  an  hour.  What  distance  will 
the  hound  run  before  he  overtakes  the  deer,  and  how  long 
will  the  chase  continue  ? 

108.  Two  men  in  New  York  hired  a  carriage  for  $25, 
to  go  to  New  Haven,  a  distance  of  t2  miles,  and  return, 
with  the  privilege  of  taking  in  three  more  persons.  Having 
gone  20  miles,  they  take  in  A;  at  New  Haven  they  take 
in  B;  and  when  within  30  miles  of  New  York  they  take 
in  C.     How  much  in  equity  ought  each  man  to  pay  ? 

109.  A  boy  went  to  a  store  and  spent  |  his  money, 
and  I  of  a  cent  more  for  pine-apples;  he  then  went  to 
another  store  and  spent  |  the  money  he  had  remaining, 
and  ^  of  a  cent  more  for  oranges;  he  now  went  to  a  third 
store  and  spent  half  the  money  he  had  remaining,  and  ^  of  a 
cent  more  for  lemons;  and  then  had  only  9  cents  remain- 
ing. How  much  money  had  he  at  first,  and  how  much  did 
he  expend  for  pine-apples,  oranges,  and  lemons  respectively  ? 

110  A  father  left  his  four  sons,  whose  ages  are  15,  11, 
8,  and  6  years  respectively,  $57 7 T,  to  be  so  divided  that 
the  respective  parts  being  placed  out,  at  6  per  cent,  simple 
interest,  shall  amount  to  equal  sums  when  they  become  21 
years  of  age.     What  are  these  parts  ? 

111.  An  individual,  at  a  public-house  borrowed  as  much 
money  as  he  had,  and  spent  12^  cents;  he  then  went  to 
another,  where  he  borrowed  as -much  money  as  he  then 
had,  and  spent  12^  cents;  then  went  to  a  third,  and  a 
fourth  and  did  the  same;  and  then  had  no  money  remain- 
ing.    How  much  money  had  he  at  first  ? 

112.  An  estate  of  $17768  is  to  be  divided  among  a 
widow,  two  sons,  and  two  daughters,  so  that  each  sou  shall 
receive  twice  as  much  as  each  daughter,  lacking  $240  ; 
and  the  widow  as  much  as  all  the  children,  lacking  $520. 
What  was  the  share  of  each  ? 

113.  A,  B,  and  C  can  perform  a  certain  piece  of  work 
in  24  days;  how  long  will  it  take  each  to  perform  the 
work  alone,  if  A  does  1|  times  as  much  as  B,  and  B  does 
I  as  much  as  C  ? 

114.  A  farmer,  having  sheep  in  two  dififerent  fields, 
sold  ]  of  the  number  from  each  field,  and  had  only  280 


288  MISCELLANEOUS    QUESTIONS.  [CHAP.  XI. 

sheep  remaining.  Kow  20  sheep  jumped  from  the  first  field 
into  the  second;  then  the  number  remaining  in  the  first 
field  was  to  the  number  remaining  in  the  second  field  as 
5  to  9,     How  many  sheep  were  there  in  each  field  at  first  ? 

115.  A  farmer  paid  five  laborers  a  certain  sum  of 
money  every  month;  to  the  first  he  paid  J  the  whole  sum, 
lacking  $16;  to  the  second  ^  of  the  remainder,  lacking  $8; 
the  third  ^  of  the  remainder,  lacking  $4;  to  the  fourth  ^ 
of  the  remainder,  lacking  $2;  and  to  the  fifth  the  remain- 
der, which  was  $11.  How  much  did  he  give  them  all 
a  month,  and  how  much  to  each  ? 

116.  A  Californian  on  his  way  home  with  $4000,  was 
met  by  a  party  that  robbed  him  of  |  of  I  of  all  he  had  ; 
a  second  party  met  and  robbed  him  of  f  of  f  of  the  re- 
mainder ;  a  third  party  met  him  and  robbed  him  of  y\  of 
^}  of  what  he  had  left  ;  and  a  fourth  party  took  from  him 
i  of  f  of  what  still  remained.  How  much  money  had  he 
left  ? 

lit.  A  gentleman  promised  his  son  a  new  arithmetic, 
if  he  would  go  to  a  certain  orchard,  which  was  entered 
through  three  gates,  and  get  such  a  number  of  apples,  that, 
on  his  return,  he  could  leave  at  the  firs*  gate,  ^  the 
apples  he  had  and  ^  an  apple  more;  at  the  second  gate, 
i  of  what  he  had  remaining  and  }  an  apple  more;  and  at 
the  third  gate,  ^  the  apples  he  still  had  remaining,  and 
i  an  apple  more,  without  cutting  any;  and  then  have 
1*1  apples  remaining.  How  many  apples  must  he  get,  and 
how  many  will  he  leave  at  the  gates,  respectively  ? 

118.  Three  men,  A,  B,  and  C,  agree  to  do  a  certain 
piece  of  work  for  $52"90;  A  and  B  calculate  that  they 
can  do  f  of  the  work;  A  and  C  calculate  that  they  can 
do  yV  <^f  the  work;  and  B  and  C  ^f.  They  are  to  be 
paid  proportionately,  to  these  estimates.  How  much 
should  each  receive  ? 

119.  The  stock  of  a  certain  bank  is  divided,  into  32 
shares,  and  is  owned  equally  by  eight  persons.  A,  B,  C, 
D,  &c.  A  sells  3  of  his  shares  to  a  ninth  person,  and  B 
sells  2  of  his  shares  to  the  Company.  What  proportiou 
of  the  whole  stock  does  A  and  B  respectively  still  own  ? 


ART.  268.]  MISCELLANEOUS   QUESTIONS.  289 

120.  A  boy,  being  asked  how  many  eggs  he  was  carry- 
ing to  market,  replied,  I  do  not  know;  but  father  said,  if 
I  had  1  dozen  more,  and  should  multiply  this  number  by 
2,  and  add  to  the  product  2  dozen ;  and  then  sell  them 
all,  at  12^  cents  a  dozen,  I  would  receive  for  them  $1'50. 
How  many  eggs  had  he  ? 

121.  A  farmer,  being  asked  how  many  sheep  he  had, 
replied,  that  he  had  them  in  four  different  fields;  and  that 
I  of  the  number  in  the  second  field  equalled  |  of  the  num- 
ber in  the  first;  f  of  the  number  in  the  second,  equalled 
I  of  the  number  in  the  third ;  and  |  of  the  number  in  the 
third,  equalled  |  of  the  number  in  the  fourth.  How  many 
sheep  in  each  field,  providing  there  are  64  more  sheep  in 
the  third  field  than  in  the  fourth,  and  how  many  in  all  ? 

122.  A  boy,  having  some  oranges,  sold  to  one  person 
1  of  all  he  had  and  10  oranges  more;  to  another,  i  of 
the  remainder  and  10  more;  to  a  third,  j%  of  what  then 
remained  and  t,  more;  to  a  fourth,  |  of  what  then 
remained  and  2  more;  to  a  fifth,  |  of  what  still  remained 
and  10  more;  and  to  the  sixth,  the  remainder.  How 
many  oranges  had  he  at  first,  and  how  many  did  he  sell 
to  each  individual,  providing  the  fifth  bought  12  oranges 
more  than  the  sixth  ? 

123.  The  interest  of  A's,  B's,  and  C's  fortune  for  nine 
years  and  4  months,  at  3  per  cent.,  is  $30380,  What  is 
the  fortune  of  each,  providing  f  of  A's  fortune  equals  | 
of  B's,  and  f  of  B's  equals  f  of  C's  ? 

124.  There  is  a  rectangular  box,  16  feet  long,  4  feet 
wide,  and  3  feet  deep.  What  must  be  the  length  and 
width  of  another  rectangular  box  of  the  same  depth,  that 
shall  contain  5625  solid  feet,  providing  its  length  and 
width  are  in  the  same  proportion  ? 

125.  A  man  at  his  death,  having  a  daughter  in  France, 
and  a  son  in  Russia,  willed,  if  the  daughter  returned,  and 
not  the  son,"  that  the  widow  should  have  |  of  the  estate  ; 
and  if  the  son  returned,  and  not  the  daughter,  that  the 
widow  should  receive  |  of  the  estate.  They  both  returned. 
How  much,  according  to  the  will,  should  each  receive, 
providing  the  estate  amounted  to  $7600  ? 

13 


290  MISCELLANEOUS    QUESTIONS.  [cHAP,  XI. 

126.  A,  B,  and  C  agree  to  do  a  certain  piece  of  work 
for  $87*87  ;  A  and  B  can  do  the  work  in  6|  days  ;  B  and 
C  in  12  days  ;  and  A  and  C  in  10  jiays.  How  much 
should  each  receive,  according  to  the  above  estimates  ? 

127.  What  will  be  the  dimensions  of  a  rectangular  box, 
which  shall  contain  4037250  solid  inches  ;  the  length, 
breadth,  and  depth  being  proportional  to  the  numbers  7, 
3,  and  2  ? 

128.  A  thief  stole  a  horse  from  a  farmer,  B,  and  made 
off  with  it ;  5  days  after,  B  got  intelligence  of  the  direc- 
tion the  thief  took,  and  followed  him  at  the  rate  of  60 
miles  a  day  ;  and  by  so  doing  gained  20  per  cent,  on  the 
thief.  At  what  rate  did  the  thief  travel  ;  how  far  must 
B  ride  before  he  overtakes  him  ;  and  how  many  days  will 
it  require. 

129.  A  drover  being  asked  how  many  animals  he  had, 
replied,  that  f  of  the  number  were  sheep  ;  |  of  the  re- 
mainder were  hogs  ;  and  what  then  rena*kined  were  calves  ; 
and  that,  if  he  should  sell  the  sheep  at  $2|  a  head ;  his 
hogs  at  $3i  ;  and  his  calves  at  $5  a  head,  he  should  re- 
ceive $519,  which  was  $119  more  than  they  cpst.  How 
many  sheep,  hogs,  and  calves  had  he,  respectively  ? 

130.  If  14  oxen  eat  2  acres  of  grass  in  3  weeks,  and  16 
oxen  -eat  6  acres  in  9  weeks,  how  many  oxen  would  eat 
24  acres  in  6  weeks  ;  the  grass  being  at  first  equal  on 
every  acre,  and  growing  uniformly  ? 

131.  If  8  oxen  eat  2  acres  of  grass  in  8  weeks  ;  and 
15  oxen  eat  5-acres  in  6  weeks  ;  for  how  many  weeks  can 
15  oxen  graze  on  6  acres,  ^e  grass  growing  uniformly  ? 

132.  If  3  acres  of  grass,  together  with  what  grew  on 
the  3  acres  whilq^they  were  grazing,  keep  13  oxen  9  weeks, 
and  in  like  manner,  4  acres  keep  20  oxen  6  weeks,  how 
many  acres  will  be  required  to  keep  36  oxen  4  weeks  ? 

133.  A  general  drew  up  his  regiment  in  the  form  of  a 
square  and  had  94  men  remaining;  soon  after  a  detach- 
ment of  485  men  more  joined  him,  whereby  he  was  enabled 
to  increase  the  side  of  the  square  by  3  men.  How  many 
soldiers  had  he  at  first  ? 

134    A  market-woman  carried  some  butter,  strawberries 


A.RT.    268.]  MISCELLANEOUS    QUESTIONS.  291 

and  eggs,  to  market ;  she  sold  her  butter,  at  25  cents  a 
pound  ;  her  strawberries  at  20  cents  a  quart ;  and  her 
eggs,  at  15  cents  a  dozen;  the  whole  amounted  to  $7"65. 
The  number  of  pounds  of  butter  equalled  the  number  of 
dozens  of  eggs  inci*eased  by  the  number  of  quarts  of  straw- 
berries ;  and  the  number  of  pounds  of  butter  increased  by 
the  number  of  quarts  of  strawberries,  or  the  number  of 
dozens  of  eggs,  would  equal  3  times  as  much  as  the  remain- 
ing number.     What  was  the  quantity  of  each  article  ? 

135.  A,  B,  C,  and  D  agree  to  a  certain  piece  of  work, 
for  $945;  A,  B,  and  C  can  perform  the  work  in  84  days; 

A,  B,  and  D,  in  72  days;  A,  C,  and  D,  in  63  days;  and 

B,  C,  and  D,  in  56  days.  How  much  money  should  each 
receive,  providing  they  all  work  until  the  work  is  com- 
plete ?      ' 

136.  A,  B,  C,  and  D  play  cards  on  this  condition:  that 
he  who  loses  shall  give  to  all  the  others  as  much  as  they 
already  have.  First  A  lost,  then  B,  then  C,  and  then  D. 
When  they  began  to  play  they  had  $162,  $82,  $42,  and 
$22,  respectively;  how  much  had  each  at  the  end  of  the 
fourth  game  ?  Suppose,  when  they  had  all  lost  in  turn, 
that  each  had  the  same  sum  of  money  $96;  how  much  had 
each  when  they  commenced  to  play  ? 

137.  For  three  successive  years ~a  merchant,  annually, 
contributed  $150  for  charitable  purposes,  and  added  yearly 
to  that  part  of  his  capital  not  thus  expended,  a  sum  equal 
its  i.  At  the  end  of  the  third  year  his  original  capital 
was  doubled.     What  was  his  capital  ? 

138.  There  is  an  island  26|  miles  in  circumference,  and 
three  men  A,  B,  and  C,  start  from  the  same  point,  and 
travel  in  the  same  direction  around  it;  A  goes  2^  miles 
an  hour;  B  goes  8^  miles  an  hour;  and  C  goes  9f  miles 
an  hour.  In  what  time  will  they  all  first  be  together;  and 
when  will  they  all  be  together  at  the  place  from  which 
they  started  ? 

139.  Three  carpenters,  A,  B,  a^^d  C,  receive  $26  for  a 
certain  amount  of  labor; — f  of  the  number  of  days  B 
labored  equaled  |  of  the  number  of  days  A  labored,  and 
I  of  the  number  of  days  C  labored  equaled  |  of  the  num* 


292  MISCELLANEOUS    QUESTIONS.  [cHAP.    XI. 

ber  of  days  B  labored;  and  A  labored  as  many  days  as  C, 
lacking  5.  How  many  days  did  each  work,  and  how 
muct  did  each  receive  a  day,  providing  -i  of  A's  daily 
wages  equaled. I  of  B's,  and  |  of  C's  equaled  |-  of  B's  ? 

140.  A  and  B  paid  $90  for  12  acres  of  pasture  for  8 
weeks,  with  an  understanding  that  B  should  have  the 
grass  that  was  then  on  the  field;  and  A,  what  grew  during 
the  time  they  were  grazing.  How  many  oxen  according 
to  the  above  understanding  can  each  turn  into  the  pasture, 
and  how  much  should  each  pay,  providing  4  acres  of  pas- 
ture, together  with  what  grew  during  the  time  they  were 
grazing,  will  keep  12  oxen  six  weeks;  and  in  a  similar 
manner,  5  acres  will  keep  35  oxen  2  weeks  ? 

141.  A  gentleman  has  in  one  bank  a  certain  number 
of  20,  15,  and  10  dollar  bills;  in  another  a  certain  num- 
ber of  5,  and  2^  dollar  gold  coins.  The  number  of  bills 
and  coins  in  both  banks  equal  3224.  How  many  of  each 
has  he,  providing  |  of  the  number  of  20  dollar  bills  equal 
I  of  the  number  of  15  dollar  bills,  |  of  the  number  of  15 
dollar  bills  equal  |  of  the  number  of  10  dollar  bills,  and 
I  of  the  number  of  5  dollar  gold  coins  are  48  more  than 
I  of  the  number  of  2|  dollar  coins;  also,  that  -f  of  the 
number  of  bills  equal  f  of  the  number  of  coins;  and 
what  amount  of  money  has  he  in  both  banks  ? 

142.  Divide  a  bar  of  lead  weighing  40  pounds  into  four 
pieces,  with  which  (and  a  pair  of  scales)  any  number  of 
pounds  from  1  to  40  may  be  weighed. 

143.  Find  the  least  possible  whole  number  which  being 
divided  by  28,  shall  leave  19  for  a  remainder;  and  being 
divided  by  19,  shall  leave  16  for  a  remainder;  and  being 
divided  by  15,>  shall  leave  11  for  a  remainder  ? 


THE   ENP. 


CONTENTS 


CHAPTER  I. 

P.«. 

IWTRODUCTORY  DEFINITIONS. NO- 

TATION.  NUMERATION, 

6 

Notation, 

6 

Roman  Table, .... 

6 

Arabic  Notation,     . 

7 

Numeration — Simple  and  Local 

Values  of  Figures,       . 

8 

Table 

9 

Do.  continued,      , 

10 

Do.                          ... 

11 

French  Method  of  Numeration 

12 

Exerci-ses  in  Numeration, 

12 

Exercises  in  Notation,    . 

13 

Fundamental  Rules  or  Arith- 

metic,     

13 

CHAPTER  II. 

ADDITION. SUBTRACTION. MUL- 

TIPLICATION.  DIVISION. 

Addition,      .         .         .  •      . 

14 

Practical  Questions, 

15 

Practical  Questions, 

19 

Subtraction 

22 

Practical  Questions, 

23 

Practical  Questions, 

27 

Practical  Questions,  combin- 

ing Addition  and  Subtrac- 

tion,       

30 

Multiplication, 

32 

iMultiplication  Table,      . 

33 

Practical  Questions, 

35 

Practical  Questions, 

37 

Practical  Questions,  combin- 

ing  Addition,   Subtraction, 

and  Multiplication, 

41 

Division, 

43 

Division  Table, 

44 

Short  Division,   . 

44 

Practical  Questions, 

46 

Long  Division,     .... 

47 

Practical  Questions, 

49 

Abstract    Examples     in     the 

Fundamental  Rules,     . 

52 

Practical  Questions  compris- 

ing the  Four  Fundamental 

Rules 

03 

CHAPTER  III. 


P«ff« 


Tables  of  Monev,  "Weights  and 

Measurrs.  —  Addition.  — 

Subtraction. — Multiplica- 

tion. AND  Division  op  Poly- 

nomials,    or      Denominate 

Numbers,      .... 

56 

Table  of  United  States  Cur- 

rency 

66 

English  or  Sterling  Monet,     . 

67 

Table 

67 

Troy  Weight, 

67 

Table 

67 

Avoirdupois  "Weight,      . 

67 

Table,       .         .         .        .        . 

67 

Apothecaries'  "Weight, 

68 

Table,       .        ...        . 

68 

Cloth  Measure, 

68 

Table, 

68 

'  Lon^  Measure, 

68 

Table,       ..... 

68 

Superficial,  or  Square  Measure, 

69 

Table 

69 

Surveyors'  Measure, 

60 

Table, 

60 

Solid,  or  Cubic  Measure, 

60 

Table 

60 

Wine  Measure, 

61 

Table, 

61 

Ale,  or  Beer  Measure,     . 

61 

Table, 

61 

Dry  Measure,  .... 

61 

Table, 

61 

Circular  Measure,  . 

62 

Table,       .  -     . 

63 

Measure  of  Time,    . 

62 

Table 

65 

Table,  exhibiting  the  number 
of  days  from  any  day  of  one 
month  to  the  same  day  of 
any  other  month  in  the 
same  year,    .... 

Books, 

Miscellaneous  Table, 

Addition  of  Denominate  Num. 
bers,      ..... 


CfNTENTS. 


Page 


Troy  "Weight,  . 

66 

Avoirdupois  "Weight, 

66 

Apothecaries'  "Weight, 

66 

Cloth  Measure, 

67 

Long  Measure, 

67 

Superficial,  or    Square  Mea 

sure,      .... 

67 

Surveyors'  Measure, 

67 

Solid,  or  Cubic  Measure, 

68 

Wine  Measure, 

68 

Ale,  or  Beer  Measure,    . 

68 

Dry  Measure,  . 

68 

Circular  Measure,  . 

69 

Measure  ol  Time,     . 

'      69 

Subtraction    of    Denominate 

Numbers, 

69 

Practical  Questions  in  Addi 

tion  and  Subtraction  of  De 

nominate  Numbers,     . 

71 

Multiplication  of  Denominate 

Numbers,      .         .         .         . 

72 

Division  of  Denominate  Num- 

bers, 

74 

Practical  Questions  combin- 

ing  Addition,  Subtraction 

Multiplication     and     Divi- 

sion of  Denominate  Num- 

bers,        

76 

Reduction, 

78 

Reduction  Descending,  .        , 

78 

Reduction  Ascending,  . 

79 

CHArTER  IV. 


Peculiar     Property     of     the 

Number  9,    , 
Multiplication    of    Abstract 

Polynomials, 
Miscellaneous  Definitions, 
Prime  Numbers, 
Table  of  Prime  Numbers, 
Resolution  of  Composite  Num 

bers  into  their  Prime  Fac 

tors,       .... 
Divisors  or  Measures  of  Num 

bers,      .        .        ,        .        , 
Common  Measure  or  Divisor, 
Greatest  Common  Measure, 
Practical  Questions  in  Com 

mon  Measure, 
Multiples, 
Practical  Questions  in  Com 

mon  Multiple, 
Abbreviated    Operations    in 

Arithmetical  Calculations, 
Examples  in  Abbreviated  Mul 

ti])lication,    .... 
Properties  or  Numbers,  . 


87 


CHAPTER  V. 


P«g« 


Fractions,   .... 
Common  Fractions, 
Reduction  of  Common  Frac 

tions 

Propositions,     .         ,         . 
Multiplication    of   Fractions 

by  Integers, 
Divisions  of  Fractions  by  In 

tegers,  .... 
Cancellation,   . 
A  Common  Denominator, 
The  Least  Common  Denomi 

nator,    .... 
Addition  of  Common   Frac 

tions,     .... 
Subtraction  of  Common  Fraa 

tions,     .... 
Multiplication    of    Common 

Fractions,     ... 
Practical  Questions  in  Mul 

tiplication  of  Fractions, 
Division    of    Common    Frac 

tions,     .... 
Practical  Questions  in  Divi 

sion  of  Fractions, 
Complex  Fractions, 
Least    Common    Multiple  of 

Fractions,     .      "  . 
Practical  Questions  in  Mul- 
tiples,   .        .        . 
Practical  Questions  in  Frac 

tions,     .... 

CHAPTER  VI 

Decimal  Fractions,     . 

Numeration  of  Decimal  Frac 
tions,     .... 

United  States  Currency,  or 
Federal  Money, 

Table  of  United  States  Cur- 
rency,   .... 

Reduction  of  Decimals  to  Com- 
mon Fractions,     . 

Reduction  of  Common  Frac 
tions'to  Decimals. 

Reduction  of  Mixed  Decimals 
to  Simple  Decimals,     . 

Repetends, 

Compound  Repetends,    . 

Additional  of  Decimals  and 
United  States  Currency, 

Practical  Questions, 

Subtraction  of  Decimals  and 
the  United  States  Currency, 

Practical  Questions, 

Multii)lication  of  Decimals 
and  the  United  States  Cur- 
rency,     


130 
130 


131 


rONTENTS. 


m 


Practical  Questions, 
Division  of  Decimals  and  the 

United  States  Currency, 
Practical  Questions, 
Practical  Questions  in  Deci 

mals  and  the  United  States 

Currency, 
Practical  Questions, 
Reduction    of   Denominate 

Fractions, 
Addition  of  Denominate  Frac- 
tions,    .... 
Practical  Questions, 
Duodecimals, 
Table,       .... 
Addition  and  Subtraction  of 

Duodecimals, 
Multiplication    of    Duodeci 

mals 

Reduction  of  Currencies, 
Table,       .... 

Table 

Aliquot  Parts, 
Analysis  by  Aliquot  Parts 
Cancellation,    . 
Ajaalysis  by  Cancellation, 

CHAPTER  VII. 


Ratio,  .... 

Proportion, 

Propositions,    . 
Simple  Proportion,     . 
Compound  Proportion, 
Conjoined  Proportion, 
Copartnership,  . 
Compound  Copartnershif, 
Alligation  Medial,   . 
Alligation  Alternate,     . 

CHAPTER  VIII. 

Percentage, 

Insurance,        ... 

Stocks,  Brokerage  and  Com 
mission, 

Custom  House  Business, 

Assessment  of  Taxes, 

Profit  and  Loss, 

Practical  Questions  in  Profit 
and  Loss, 
Simple  Interest, 


Page 
132 


133 
134 


135 
140 

140 

143 
144 
146 

146 

147 

147 
160 
160 
150 
152 
152 
153 
153 


154 
155 
156 
156 
162 
164 
165 
169 
170 
171 


175 
176 

177 
179 
181 
183 

185 
18« 


V»9> 

Table,  showinjf  the  Aliquot 

Parts  of  a  Year  or  Month,   .  193 

Problems  in  Interest,       .        .  19« 

Discount,         ....  197 

Partial  Payments,    ,        .        ,  199 

Compound  Interest,    .         .         .  201 

Banking  and  Notes,        .        .  202 

Bank  Discount,        .        .        .  204 

Average,  .         .         *        .  206 

Mercantile  Calculations. 

Equation  of  Payments,  .        .  206 

Trade  and  Barter,    .        .        .  217 

CHAPTER  IX. 

Progression,         .         .         .  221 

Arithmetical  Progression,      .  221 

Geometrical  Progression,       .  224 

CHAPTER  X. 

Involution  and  Evolution,        .  227 

Involution,       ....  227 

Evolution,         ....  228 

Square  Root,       ....  229 
Mechanical    Application    of 

the  Foregoing,      .        .        .  233 

Length  of  Braces,  .        .        .  238 

Length  of  Rafters,  .        .  239 

Cube  Root,  .         .        .         .241- 

Practical  Questions  in  Cube 

Root, 261 

Guaging,  ....  262 

CHAPTER  XI. 

Mensuration 254 

Geometrical  Definations,  .  254 
Plane  Figures,  .  .  .  264 
Solid  Figures,  .  .  .  .256 
Mensuration  of  Surfaces,  &c.  257 
Mensuration  of  Solids,  .  .  263 
Table  of  Multiples  for  Me- 
chanics   265 

The  Five  Regular  Bodies,      .  267 
Surfaces  of  the  Five  Regular 

Bodies 268 

Table,     ' 269 

Solid  ity  of  the  Regular  Bodies,  270 

Table, 271 

PHiLoscrHicAL  Problems,  .        .  271 

Miscellaneous  Questions,        .  274 


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A    CHRONOLOGICAL    SCHOOL    HISTORY    OF    THE    UNITED 

STATES,  illustrated  by  painted  Plates  of  the  four  last  Centuries,  prepared  on  the  prin- 
ciple of  Bems  Chaet of  UifiTEESAL  HisTOET, by  M188  Elizabeth  P.  Peabodt.  1  voL 
12mo.    Price 

The  publishers  would  invite  the  attention  of  all  wishing  to  commence  classes  in  the 
history  of  the  United  States,  to  the  following  flattering  commendation,  given  after  having 
read  the  work  in  manuscript,  by  Prof.  J.  H.  EAYMOND,  LL.D.,  late  of  the  Univeesity 
OF  K0CHE8TEE,  and  now  Principal  of  the  Polytechnic  School,  Brooklyn. 

"  It  aflfords  me  peculiar  satisfaction  to  learn  that  Miss  Peabody  has  undertaken  to  pre- 
pare a  work  on  the  history  of  the  United  States  for  the  use  of  schools.  I  certainly  know 
of  none  who  combines  in  such  large  measure,  the  rare  talents  and  acquirements,  both 
natural  and  moral,  which  such  an  undertaking  requires.  The  chronological  method  of 
Bern,  which  she  incorporates  in  her  plan,  I  have  long  regarded  as  OUT  OF  SIGHT 
SUPERIOK  to  any  other  scheme  of  GhranologicaZ  Mnemonica  e/ver  invented.  I  think 
you  cannot  do  a  better  thing  for  schools— I  should  also  hope  for  yourselves — than  to  put 
it  in  type." 

The  venerable  Dr.  NOTT,  of  Uniok  College,  having  also  examined  the  manuscript, 
and  expressed  his  cordial  approbation  of  the  history,  adds, — "The  plan  of  this  work  is 
calculated  to  excite  and  sustain  the  imagination,  not  merely  by  appealing  to  the  eye,  in 
Impressing  its  chronology,  but  also  by  a  graphic  outline  of  the  history  of  each  Colony, 
and  of  the  Federal  Union,  in  such  a  manner  as  to  preserve  their  respective  individuali- 
ties and  peculiar  spirit." 

W&  are  also  prepared  to  furnish 

BEM'S  CHARTS  OF  UNIVERSAL  HISTORY,  with  the  Manual  pre- 
pared by  Miss  Peabody,  at  the  instigation  of  Dr.  Barnas  Seabs,  late  Secretary  of  tho 
Massachusetts  Board  of  Education,  and  now  President  of  Brown  University. 

And  we  have  in  our  possession,  manuscript  letters  in  testimony  of  its  value,  from 

Dr.  NOTT,  and  Professors  Newman  Hicook  and  Taylor  Lewis,  of  Union  College 
Prof.  Raymoxd,  late  of  the  Rochester  University,  Professors  Andrews  and  Kingslbt, 
of  Marietta  College,  Prof.  Gregory,  of  Detroit,  now  editor  of  the  Michigan  Journal  of 
Ednication,  Rev.  Eban  S.  Stearns,  late  Principal  of  the  Normal  School  at  West  Newton, 
Mass.,  Kev.  F.  A.  Adams,  of  Orange,  N.  J.,  Prof.  Burton,  then  of  Girard  College,  Mr. 
Alonzo  Crittenden,  of  Packer  Institute,  Brooklyn,  Dr.  Isaac  Ferris.  Chancellor  of  the 
University  of  New  York,  Dr.  J.  Romayn  Beck,  late  of  Albany,  Dr.  W.  B.  Sprague,  of 
Albany,  and  many  others  who  have  used  it,  especially  ladies  of  the  first  class  of  teachers. 

HISTORICAL  &  CHRONOLOGICAL  TABLES  :  for  use  in  Elementary 
Instruction  in  HISTORY.  By  Dr.  Charles  Peteb,  Director  of  the  Gymnasium  in 
Auckland.    Translated  from  the  German  (3d  Ed.). 

Prefatory  Note. 

It  would  be  diflScult  to  compress  within  a  smaller  space  and  in  a  more  convenient  form, 
the  amount  of  historical  information  that  is  comprised  in  the  following  pages.  They  are 
prepared  by  a  distinguished  German  scholar,  who  has  great  experience  in  the  preparation 
of  larger  and  smaller  works  for  Schools  on  history.  This  little  manual  is  translated  in 
the  hope  and  belief  that  it  will  prove  extensively  useful  as  an  outline  and  resume  of  his- 
torical facts  in  their  chronological  connections,  in  both  higher  and  lower  seminaries  of 
Instruction  in  America,  and  also  an  excellent  companion  to  private  students  in  history. 

A.  C.  KENDEICK, 

UjaVBBfllTY  OF  ROOHBSTSE. 


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